boolean algebra

Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of conjunction x∧y, disjunction x∨y, and complement ¬x. The Boolean operations are these and all other operations that can be built from these, such as x∧(y∨z). These turn out to coincide with the set of all operations on the set {0,1} that take only finitely many arguments; there are 22n such operations when there are n arguments.

The laws of Boolean algebra can be defined axiomatically as certain equations called axioms together with their logical consequences called theorems, or semantically as those equations that are true for every possible assignment of 0 or 1 to their variables. The axiomatic approach is sound and complete in the sense that it proves respectively neither more nor fewer laws than the semantic approach.

Values

Boolean algebra is the algebra of two values. These are usually taken to be 0 and 1, as we shall do here, although F and T, false and true, etc. are also in common use. For the purpose of understanding Boolean algebra any Boolean domain of two values will do.

Regardless of nomenclature, the values are customarily thought of as essentially logical in character and are therefore referred to as truth values, in contrast to the natural numbers or the reals which are considered numerical values. On the other hand the algebra of the integers modulo 2, while ostensibly just as numeric as the integers themselves, was shown to constitute exactly Boolean algebra, originally by I.I. Zhegalkin in 1927 and rediscovered independently in the west by Marshall Stone in 1936. So in fact there is some ambiguity in the true nature of Boolean algebra: it can be viewed as either logical or numeric in character.

More generally Boolean algebra is the algebra of values from any Boolean algebra as a model of the laws of Boolean algebra. For example the bit vectors of a given length, as with say 32-bit computer words, can be combined with Boolean operations in the same way as individual bits, thereby forming a 232-element Boolean algebra under those operations. Any such combination applies the same Boolean operation to all bits simultaneously. This passage from the Boolean algebra of 0 and 1 to these more general Boolean algebras is the Boolean counterpart of the passage from the algebra of the ring of integers to the algebra of commutative rings in general. The two-element Boolean algebra is the prototypical Boolean algebra in the same sense as the ring of integers is the prototypical commutative ring. Boolean logic as the subject matter of this article is independent of the choice of Boolean algebra (the same equations hold of every nontrivial Boolean algebra); hence, there is no need here to consider any Boolean algebra other than the two-element one. The article on Boolean algebra (structure) treats Boolean algebras themselves.
operations

Basic operations

After values, the next ingredient of any algebraic system is its operations. Whereas elementary algebra is based on numeric operations multiplication xy, addition x + y, and negation −x, Boolean algebra is customarily based on logical counterparts to those operations, namely conjunction x∧y (AND), disjunction x∨y (OR), and complement or negation ¬x (NOT). In electronics, the AND is represented as a multiplication, the OR is represented as an addition, and the NOT is represented with an overbar: x ∧ y and x ∨ y, therefore, become xy and x + y.
Conjunction is the closest of these three to its numerical counterpart, in fact on 0 and 1 it is multiplication. As a logical operation the conjunction of two propositions is true when both propositions are true, and otherwise is false. The first column of Figure 1 below tabulates the values of x∧y for the four possible valuations for x and y; such a tabulation is traditionally called a truth table.
Disjunction, in the second column of the figures, works almost like addition, with one exception: the disjunction of 1 and 1 is neither 2 nor 0 but 1. Thus the disjunction of two propositions is false when both propositions are false, and otherwise is true. This is just the definition of conjunction with true and false interchanged everywhere; because of this we say that disjunction is the dual of conjunction.
Logical negation however does not work like numerical negation at all. Instead it corresponds to incrementation: ¬x = x+1 mod 2. Yet it shares in common with numerical negation the property that applying it twice returns the original value: ¬¬x = x, just as −(−x) = x. An operation with this property is called an involution. The set {0,1} has two permutations, both involutary, namely the identity, no movement, corresponding to numerical negation mod 2 (since +1 = −1 mod 2), and SWAP, corresponding to logical negation. Using negation we can formalize the notion that conjunction is dual to disjunction via De Morgan's laws, ¬(x∧y) = ¬x ∨ ¬y and ¬(x∨y) = ¬x ∧ ¬y. These can also be construed as definitions of conjunction in terms of disjunction and vice versa: x∧y = ¬(¬x ∨ ¬y) and x∨y = ¬(¬x ∧ ¬y).

Various representations of Boolean operations

Figure 2 shows the symbols used in digital electronics for conjunction and disjunction; the input ports are on the left and the signals flow through to the output port on the right. Inverters negating the input signals on the way in, or the output signals on the way out, are represented as circles on the port to be inverted.

Derived operations

Other Boolean operations are derivable from these by composition. For example implication x→y (IMP), in the third column of the figures, is a binary operation which is false when x is true and y is false, and true otherwise. It can be expressed as x→y = ¬x∨y (the OR-gate of Figure 2 with the x input inverted), or equivalently ¬(x∧¬y) (its De Morgan equivalent in Figure 3). In logic this operation is called material implication, to distinguish it from related but non-Boolean logical concepts such as entailment and relevant implication. The idea is that an implication x→y is by default true (the weaker truth value in the sense that false implies true but not vice versa) unless its premise or antecedent x holds, in which case the truth of the implication is that of its conclusion or consequent y.
Although disjunction is not the exact counterpart of numerical addition, Boolean algebra nonetheless does have an exact counterpart, called exclusive-or (XOR) or parity, x⊕y. As shown in the fourth column of the figures, the exclusive-or of two propositions is true just when exactly one of the propositions is true; equivalently when an odd number of the propositions is true, whence the name "parity". Exclusive-or is the operation of addition mod 2. The exclusive-or of any value with itself vanishes, x⊕x = 0, since the arguments have an even number of whatever value x has. Its digital electronics symbol is shown in Figure 2, being a hybrid of the disjunction symbol and the equality symbol. The latter reflects the fact that the negation (which is also the dual) of XOR, ¬(x⊕y), is logical equivalence, EQV, being true just when x and y are equal, either both true or both false. XOR and EQV are the only binary Boolean operations that are commutative and whose truth tables have equally many 0s and 1s. Exclusive-or together with conjunction constitute yet another complete basis for Boolean algebra, with the Boolean operations reformulated as the Zhegalkin polynomials.
Another example is Sheffer stroke, x|y, the NAND gate in digital electronics, which is false when both arguments are true, and true otherwise. NAND is definable by composition of negation with conjunction as x |y = ¬(x∧y). It does not have its own schematic symbol as it is easily represented as an AND gate with an inverted output. Unlike conjunction and disjunction, NAND is a binary operation that can be used to obtain negation, via the definition ¬x = x|x. With negation in hand one can then in turn define conjunction in terms of NAND via x∧y = ¬(x|y), from which all other Boolean operations of nonzero arity can then be obtained. NOR, ¬(x∨y), as the evident dual of NAND serves this purpose equally well. This universal character of NAND and NOR makes them a popular choice for gate arrays, integrated circuits with multiple general-purpose gates.
The above-mentioned duality of conjunction and disjunction is exposed further by De Morgan's laws, ¬(x∧y) = ¬x∨¬y and ¬(x∨y) = ¬x∧¬y. Figure 3 illustrates De Morgan's laws by giving for each gate its De Morgan dual, converted back to the original operation with inverters on both inputs and the outputs. In the case of implication, taking the form of an OR-gate with one inverter on disjunction, that inverter is cancelled by the second inverter that would have gone there. The De Morgan dual of XOR is just XOR with an inverter on the output (there is no separate symbol); as with implication, putting inverters on all three ports cancels the dual's output inverter. More generally, changing an odd number of inverters on an XOR gate produces the dual gate, an even number leaves the gate's functionality unchanged.
As with all the other laws in this section, De Morgan's laws may be verified case by case for each of the 2ⁿ possible valuations of the n variables occurring in the law, here two variables and hence 2² = 4 valuations. De Morgan's laws play a role in putting Boolean terms in certain normal forms, one of which we will encounter later in the section on soundness and completeness.
Figure 4 illustrates the corresponding Venn diagrams for each of the four operations presented in Figures 1-3. The interior (respectively exterior) of each circle represents the value true (respectively false) for the corresponding input, x or y. The convention followed here is to represent the true or 1 outputs as dark regions and false as light, but the reverse convention is also sometimes used.

All Boolean operations

There are infinitely many expressions that can be built from two variables using the above operations, suggesting great expressiveness. Yet a straightforward counting argument shows that only 16 distinct binary operations on two values are possible. Any given binary operation is determined by its output values for each possible combination of input values. The two arguments have 2 × 2 = 4 possible combinations of values between them, and there are 2⁴ = 16 ways of assigning an output value to each of these four input values. The choice of one of these 16 assignments then determines the operation; so all together there are only 16 distinct binary operations.
The 16 binary Boolean operations can be organized as follows:
Two constant operations, 0 and 1.
Four operations dependent on one variable, namely x, ¬x, y, and ¬y, whose truth tables amount to two juxtaposed rectangles, one containing two 1s and the other two 0s.
Two operations with a "checkerboard" truth table, namely XOR and EQV.
Four operations are obtained from disjunction with some subset of its inputs negated, namely x∨y, x→y, y→x, and x|y; their truth tables contain a single 0.
The final four come from the same treatment applied to conjunction, having a single 1 in their truth tables.
10 of the 16 operations depend on both variables; all are representable schematically as an AND-gate, an OR-gate, or an XOR-gate, with one port optionally inverted. For the AND and OR gates the location of each inverter matters, for the XOR gate it does not, only whether there is an even or odd number of inverters.
Operations of other arities are possible. For example the ternary counterpart of disjunction can be obtained as (x∨y)∨z. In general an n-ary operation, one having n inputs, has 2ⁿ possible valuations of those inputs. An operation has two possibilities for each of these, whence there exist 2²ⁿ n-ary Boolean operations. For example, there are 2³² = 4,294,967,296 operations with 5 inputs.
Although Boolean algebra confines attention to operations that return a single bit, the concept generalizes to operations that take n bits in and return x bits instead of one bit. Digital circuit designers draw such operations as suitably shaped boxes with n wires entering on the left and m wires exiting on the right. Such multi-output operations can be understood simply as m n-ary operations. The operation count must then be raised to the m-th power, or, in the case of n inputs, (2²ⁿ)^m = 2^m2n operations. The number of Boolean operations of this generalized kind with say 5 inputs and 5 outputs is 1.46 × 10⁴⁸. A logic gate or computer module mapping 32 bits to 32 bits could implement any of 5.47 × 10^{41,373,247,567} operations, more than is obtained by squaring a googol 28 times.

Axioms

With values and operations in hand, the next aspect of Boolean algebra is that of laws or properties. As with many kinds of algebra, the principal laws take the form of equations between terms built up from variables using the operations of the algebra. Such an equation is deemed a law or identity just when both sides have the same value for all values of the variables, equivalently when the two terms denote the same operation.
Numeric algebra has laws such as commutativity of addition and multiplication, x + y = y + x and xy = yx. Similarly, Boolean algebra has commutativity in that x ∨ y = y ∨ x for disjunction and x ∧ y = y ∧ x for conjunction. Not all binary operations are commutative; Boolean implication is not commutative, like subtraction and division.
Another equally fundamental law is associativity, which in the case of numeric multiplication is expressed as x(yz) = (xy)z, justifying abbreviating both sides to xyz and thinking of multiplication as a single ternary operation. All four of numeric addition and multiplication and logical disjunction and conjunction are associative, giving for the latter two the Boolean laws x ∨ (y ∨ z) = (x ∨ y) ∨ z and x ∧ (y ∧ z) = (x ∧ y) ∧ z.
Again numeric subtraction and logical implication serve as examples, this time of binary operations that are not associative. On the other hand exclusive-or, being just addition mod 2, is both commutative and associative.
Boolean algebra does not completely mirror numeric algebra however, as both conjunction and disjunction satisfy idempotence, expressed respectively as x ∧ x = x and x ∨ x = x. These laws are easily verified by considering the two valuations 0 and 1 for x. But since 2 + 2 = 2 × 2 = 4 in arithmetic, clearly numeric addition and multiplication are not idempotent. With arithmetic mod 2 on the other hand, multiplication is idempotent, though not addition since 1 + 1 = 0 mod 2, reflected logically in the idempotence of conjunction but not of exclusive-or.
A more subtle difference between number and logic is with x(x + y) and x + xy, neither of which equal x numerically. In Boolean algebra however, both x ∧ (x ∨ y) and x ∨ (x ∧ y) are equal to x, as can be verified for each of the four possible valuations for x and y. These two Boolean laws are called the laws of absorption. These laws (both are needed) together with the associativity, commutativity, and idempotence of conjunction and disjunction constitute the defining laws or axioms of lattice theory. (Actually idempotence can be derived from the other axioms.)
Another law common to numbers and truth values is distributivity of multiplication over addition, when paired with distributivity of conjunction over disjunction. Numerically we have x(y + z) = xy + xz, whose Boolean algebra counterpart is x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). On the other hand Boolean algebra also has distributivity of disjunction over conjunction, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z), for which there is no numeric counterpart, consider 1 + 2 × 3 = 7 whereas (1 + 2) × (1 + 3) = 12. Like associativity, distributivity has three variables and so requires checking 2³ = 8 cases.
Either distributivity law for Boolean algebra entails the other. Adding either to the axioms for lattices axiomatizes the theory of distributive lattices. That theory does not need the idempotence axioms because they follow from the six absorption, distributivity, and associativity laws.
Two Boolean laws having no numeric counterpart are the laws characterizing logical negation, namely x ∧ ¬x = 0 and x ∨ ¬x = 1. These are the only laws thus far that have required constants. It then follows that x ∧ 0 = x ∧ (x ∧ ¬x) = (x ∧ x) ∧ ¬x = x ∧ ¬x = 0, showing that 0 works with conjunction in logic just as it does with multiplication of numbers. Also x ∨ 0 = x ∨ (x ∧ ¬x) = x by absorption. Dualizing this reasoning, we obtain x ∨ 1 = 1 and x ∧ 1 = x. Alternatively we can justify these laws more directly simply by checking them for each of the two valuations of x.
The six laws of lattice theory along with these first two laws for negation axiomatize the theory of complemented lattices. Including either distributivity law then axiomatizes the theory of complemented distributive lattices. For convenience we collect these nine laws in one place as follows.

		associativity
		commutativity
		absorption
		distributivity
		complements

The next two sections show that this theory is sufficient to axiomatize all the valid laws or identities of two-valued logic, that is, Boolean algebra. It follows that Boolean algebra as commonly defined in terms of these axioms coincides with the intuitive semantic notion of the valid identities of two-valued logic.

Derivations

While the Boolean laws enumerated in the previous section are certainly highlights of Boolean algebra, they by no means exhaust the laws, of which there are infinitely many, nor do they even exhaust the highlights. As it is out of the question to proceed in the ad hoc way of the preceding section for ever, the question arises as to how best to present the remaining laws.
One way of establishing an equation as being a law is to verify its truth for all valuations of its variables, sometimes called the method of truth tables. This is the method we depended on in the previous section to justify each law as we introduced it, constituting the semantic approach to establishing laws. From a practical standpoint the method lends itself to computer implementation for 20-30 variables because the enumeration of valuations is straightforward to program and boring to carry out, making it ideal work for a computer. Because there are 2ⁿ valuations to check the method starts to become impractical as 40 variables is approached. Beyond that the approach becomes of value mainly as the in-principle semantic definition of what constitutes an identically true or valid equation.
In contrast the syntactic approach is to derive new laws by symbolic manipulation from already established laws such as those listed in the previous section. (This is not to imply that derivations of a law shorter than the length of a semantic verification of that law need exist, although some thousand-variable laws impossible to verify by enumeration of valuations can have quite short derivations.) Here is an example showing the derivation of (w∨x)∨(y∨z) = (w∨y)∨(x∨z) from just the commutativity and associativity of disjunction.
(w∨x)∨(y∨z)
= ((w∨x)∨y)∨z
= (w∨(x∨y))∨z
= (w∨(y∨x))∨z
= ((w∨y)∨x)∨z
= (w∨y)∨(x∨z)
The first two and last two steps appealed to associativity while the middle step used commutativity.
The rules of derivation for forming new laws from old can be assumed to be those permissible in high school algebra. For definiteness however it is worthwhile formulating a well-defined set of rules showing exactly what is needed. These are the domain-independent rules of equational logic, as sound for logic as they are for numerical domains or any other kind.
Reflexivity: t = t. That is, any equation whose two sides are the same term t is a law. (While arguably an axiom rather than a rule since it has no premises, we classify it as a rule because like the other three rules it is domain-independent, making no mention of specific logical, numeric, or other operations.)
Symmetry: From s = t infer t = s. That is, the two sides of a law may be interchanged. Intuitively one attaches no importance to which side of an equation a term comes from.
Transitivity: A chain s = t = u of two laws yields the law s = u. (This law of "cutting out the middleman" is applied four times in the above example to eliminate the intermediate terms.)
Substitution: Given two laws and a variable, each occurrence of that variable in the first law may be replaced by one or the other side of the second law. (Distinct occurrences can be replaced by distinct sides, but every occurrence must be replaced by one or the other side.)
While the first equation in the above example might seem simply a straightforward application of the associativity law, when analyzed more carefully according to the above rules it can be seen to require something more. We can justify it in terms of the reflexivity and substitution rules. Beginning with the laws x∨(y∨z) = (x∨y)∨z and w∨x = w∨x, we use substitution to replace both occurrences of x by w∨x to arrive at the first equation. All five equations in the chain are accounted for along similar lines, with commutativity in place of associativity in the middle equation.

Soundness and completeness

It can be shown that the two approaches, semantic and syntactic, to constructing all the laws of Boolean algebra lead to the same set of laws. We say that the syntactic approach is sound when it yields a subset of the semantically obtained laws, and complete when it yields a superset thereof. We can then restate this coinciding of the semantic and syntactic approaches as the soundness and completeness of the syntactic approach with respect to (or as calibrated by) the semantic approach.
Soundness follows firstly from the fact that the initial laws or axioms we started from were all identities, that is, semantically true laws. Secondly it depends on the easily verified fact that the rules preserve identities.
Completeness can be proved by first deriving a few additional useful laws and then showing how to use the axioms and rules to prove that a term with n variables, ordered alphabetically say, is equal to its n-ary normal form, namely a unique term associated with the n-ary Boolean operation realized by that term with the variables in that order. It then follows that if two terms denote the same operation (the same thing as being semantically equal), they are both provably equal to the normal form term denoting that operation, and hence by transitivity provably equal to each other.
There is more than one suitable choice of normal form, but complete disjunctive normal form will do. A literal is either a variable or a negated variable. A disjunctive normal form (DNF) term is a disjunction of conjunctions of literals. (Associativity allows a term such as x∨(y∨z) to be viewed as the ternary disjunction x∨y∨z, likewise for longer disjunctions, and similarly for conjunction.) A DNF term is complete when every disjunct (conjunction) contains exactly one occurrence of each variable, independently of whether or not the variable is negated. Such a conjunction uniquely represents the operation it denotes by virtue of serving as a coding of those valuations at which the operation returns 1. Each conjunction codes the valuation setting the positively occurring variables to 1 and the negated ones to 0; the value of the conjunction at that valuation is 1, and hence so is the whole term. At valuations corresponding to omitted conjunctions, all conjunctions present in the term evaluate to 0 and hence so does the whole term.
In outline the general technique for converting any term to its normal form, or normalizing it, is to use De Morgan's laws to push the negations down to the variables. This yields monotone normal form, a term built from literals with conjunctions and disjunctions. For example ¬(x ∨ (¬y∧z)) becomes ¬x ∧ ¬(¬y∧z) and then ¬x ∧ (¬¬y∨¬z). Applying ¬¬x = x then yields ¬x ∧ (y∨¬z).
Next use distributivity of conjunction over disjunction to push all conjunctions down below all disjunctions, yielding a DNF term. This makes the above example (¬x∧y) ∨ (¬x∧¬z).
Then for each variable y, replace each conjunction x not containing y with the disjunction of two copies of x, with y conjoined to one copy of x and ¬y conjoined to the other, in the end yielding a complete DNF term. (This is one place where an auxiliary law helps, in this case x = x∧1 = x∧(y∨¬y) = (x∧y) ∨ (x∧¬y).) In the above example the first conjunction lacks z while the second lacks y; expanding appropriately yields the complete DNF term (¬x∧y∧z) ∨ (¬x∧y∧¬z) ∨ (¬x∧¬z∧y) ∨ (¬x∧¬z∧¬y).
Next use commutativity to put the literals in each conjunction in alphabetical order. The example becomes (¬x∧y∧z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧¬y∧¬z). This brings any repeated copies of literals next to each other; delete the redundant copies using idempotence of conjunction, not needed in our example.
Lastly order the disjuncts according to a suitable uniformly applied criterion. The criterion we use here is to read the positive and negative literals of a conjunction as respectively 1 and 0 bits, and to read the bits in a conjunction as a binary number. In our example the bits are 011, 010, 010, 000, or in decimal 3, 2, 2, 0. Ordering them numerically as 0, 2, 2, 3 yields (¬x∧¬y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧z). Note that these bits are exactly those valuations for x, y, and z that satisfy our original term ¬(x∨(¬y∧z)). Complete DNF amounts to a canonical way of representing the truth table for the original term as another term.
Repeated conjunctions can then be deleted using idempotence of disjunction, which simplifies our example to (¬x∧¬y∧¬z) ∨ (¬x∧y∧¬z) ∨ (¬x∧y∧z).
In this way we have proved that the term we started with is equal to the normal form term for the operation it denotes. Hence all terms denoting that operation are provably equal to the same normal form term and hence by transitivity to each other.

A Greener Apple

Apple has been criticized by some environmental organizations for not being a leader in removing toxic chemicals from its new products, and for not aggressively or properly recycling its old products. Upon investigating Apple’s current practices and progress towards these goals, I was surprised to learn that in many cases Apple is ahead of, or will soon be ahead of, most of its competitors in these areas. Whatever other improvements we need to make, it is certainly clear that we have failed to communicate the things that we are doing well.

It is generally not Apple’s policy to trumpet our plans for the future; we tend to talk about the things we have just accomplished. Unfortunately this policy has left our customers, shareholders, employees and the industry in the dark about Apple’s desires and plans to become greener. Our stakeholders deserve and expect more from us, and they’re right to do so. They want us to be a leader in this area, just as we are in the other areas of our business. So today we’re changing our policy.

Now I’d like to tell you what we are doing to remove toxic chemicals from our new products, and to more aggressively recycle our old products.
Removing Toxic Chemicals

Lead

Many of the dangerous chemicals we all want to eliminate from electronic products are found in very small amounts, but there’s one toxic substance that some companies still ship by the pound, and that’s the lead contained in their cathode-ray tube (CRT) displays. A typical CRT contains approximately 3 pounds (1.36 kg) of lead. In mid-2006, Apple became the first company in the computer industry to completely eliminate CRTs. The effect has been stunning — our first CRT-based iMac contained 484 grams of lead; our current third-generation LCD-based iMac contains less than 1 gram of lead.

Apple completely eliminated the use of CRTs in mid-2006.

A note of comparison — Dell, Gateway, Hewlett Packard and Lenovo still ship CRT displays today.
Cadmium
Hexavalent Chromium
Decabromodiphenyl Ether

The European Union is generally ahead of the U.S. in restricting toxic substances in electronic products. Their latest restrictions, known as RoHS, went into effect in July 2006. All Apple products worldwide comply with RoHS. Our manufacturing policies had already restricted or banned most of the chemicals covered by RoHS, and Apple began introducing fully RoHS-compliant products a year before the European deadline.

Almost a year later, however, some electronics companies can only claim their products are RoHS compliant because of certain little-known exemptions granted by the EU. Despite the tough restrictions of RoHS, these exemptions let companies ship electronics that still contain high concentrations of two hazardous substances — hexavalent chromium, the carcinogen against which Erin Brockovich famously campaigned, and the brominated flame retardant decabromodiphenyl ether (DecaBDE), which is also feared to have adverse health effects. Apple phased out these and many other chemicals several years ago through design innovations and the use of higher quality metals and plastics.

Apple products met both the spirit and letter of the RoHS restrictions on cadmium, hexavalent chromium and brominated flame retardants years before RoHS went into effect.

A note of comparison — Some electronics companies, whose names you know, still rely on RoHS exemptions and use these toxic chemicals in their products today.
Arsenic
Mercury

Arsenic and mercury are industry standard materials used in liquid crystal displays (LCDs). Arsenic is added during the manufacturing of the high performance glass used in LCDs to prevent the formation of defects, and the fluorescent lamps used to illuminate LCDs contain minute amounts of mercury. Apple is on track to introduce our first displays using arsenic-free glass in 2007. A small number of high performance integrated circuits (ICs) will continue to contain a minute amount of arsenic as an element of the semiconductor substrate.

To eliminate mercury in our displays, we need to transition from fluorescent lamps to light-emitting diodes (LEDs) to illuminate the displays. Fortunately, all iPod displays already use LEDs for illumination, and therefore contain no mercury. We plan to introduce our first Macs with LED backlight technology in 2007. Our ability to completely eliminate fluorescent lamps in all of our displays depends on how fast the LCD industry can transition to LED backlighting for larger displays.

Apple plans to completely eliminate the use of arsenic in all of its displays by the end of 2008.

Apple plans to reduce and eventually eliminate the use of mercury by transitioning to LED backlighting for all displays when technically and economically feasible.
Polyvinyl Chloride
Brominated flame retardants

Some companies have made promises to phase out other toxic chemicals like polyvinyl chloride (PVC), a type of plastic primarily used in the construction industry but also found in computer parts and cables, and brominated flame retardants, or BFRs, which reduce the risk of fire. Apple began phasing out PVC twelve years ago and began restricting BFRs in 2001. For the past several years, we have been developing alternative materials that can replace these chemicals without compromising the safety or quality of our products. Today, we’ve successfully eliminated the largest applications of PVC and BFRs in our products, and we’re close to eliminating these chemicals altogether. For example, more than three million iPods have already shipped with a BFR-free laminate on their logic boards.

Dell and Lenovo have publicly stated that they plan to eliminate the use of PVC and BFRs in their products in 2009. Hewlett Packard has not yet publicly stated when they will eliminate the use of PVC and BFRs in their products, but has said that they will publish a plan by the end of 2007 which will state when in the future they will eliminate the use of these toxic chemicals in their products.

Apple plans to completely eliminate the use of PVC and BFRs in its products by the end of 2008.

A note of comparison — In 2007 HP stated that they will remove PVC from all their packaging. Apple did this 12 years ago. Last year, Dell began the process of phasing out large quantities of brominated flame retardants in large plastic enclosure parts. Apple’s plastic enclosure parts have been bromine-free since 2002.

In one environmental group’s recent scorecard, Dell, HP and Lenovo all scored higher than Apple because of their plans (or “plans for releasing plans” in the case of HP). In reality, Apple is ahead of all of these companies in eliminating toxic chemicals from its products.
Recycling Our Products (E-Waste)

Apple started recycling in 1994 and today we operate recycling programs in countries where more than 82% of all Macs and iPods are sold. By the end of this year, that figure will increase to 93%. How successful are these programs?

Currently, there is no industry standard way to measure the effectiveness of a company’s recycling programs. Dell has proposed a simple measure - assume a seven year product lifetime, and measure the percentage of the total weight you recycle each year compared to the total weight of what you sold seven years earlier. This makes sense to us, and has the added advantages of clarity and simplicity.

Apple recycled 13 million pounds of e-waste in 2006, which is equal to 9.5% of the weight of all products Apple sold seven years earlier. We expect this percentage to grow to 13% in 2007, and to 20% in 2008. By 2010, we forecast recycling 19 million pounds of e-waste per year — nearly 30% of the product weight we sold seven years earlier.
Weight Recycled as % of Past Sales
Chart shows an upward trend starting with 1.5% in 2002, up to an actual 9.5% in 2006, and an estimated 28% in 2010.

A note of comparison — the latest figures from HP and Dell are each around 10% per year, and neither company has yet disclosed plans to grow this percentage in the future. By 2010, Apple may be recycling significantly more than either Dell or HP as a percentage of past sales weight.

All the e-waste we collect in North America is processed in the U.S., and nothing is shipped overseas for disposal. We carefully review “environmental fate” submissions from each vendor, so we know how raw materials are handled at the end of the recycling process. We hold our recycling vendors to the highest environmental standards in the industry. In addition to annual compliance audits, we also review the performance of their downstream vendors. They must comply with all applicable health and safety laws, and we do not allow the use of prison labor at any stage of the recycling process.

Producers must also take responsibility for the design and material choices that create the product in the first place. It is these choices that fundamentally determine the weight and recycling value of material waste at the end of a product’s life. The iMac is a world-class example of material efficiency, having shed 60% of its weight since its debut in 1998. Our designs use aircraft-grade aluminum, stainless steel and high-grade plastics that are in high demand from recyclers, who recover and resell these raw materials for use in other types of products. Few of our competitors do the same.

Let me take a moment to talk specifically about iPods, even though they are included in the above data. All of Apple’s U.S. retail stores, which now number more than 150, take back unwanted iPods for environmentally friendly disposal free of charge. As an incentive, we even offer customers a 10% discount on a new iPod when they bring their old iPod to our stores for proper disposal. This summer we’re expanding it to Apple retail stores worldwide, and we’re also extending it to include free shipping from anywhere in the U.S. No product purchases are required for any of our free take back programs. In a few months, we think we’ll have ‘best of breed’ iPod recycling programs in the U.S., and we plan to continue to expand our free iPod recycling programs globally in the future.

By 2010, Apple may be recycling significantly more than either Dell or HP as a percentage of past sales weight.

All the e-waste we collect in North America is processed in the U.S., and nothing is shipped overseas for disposal.

Apple products are designed using high quality materials that are in high demand from recyclers.
The Future

Today is the first time we have openly discussed our plans to become a greener Apple. It will not be the last. We will be providing updates of our efforts and accomplishments at least annually, most likely around this time of the year. And we plan to bring other environmental issues to the table as well, such as the energy efficiency of the products in our industry. We are also beginning to explore the overall carbon “footprint” of our products, and may have some interesting data and issues to share later this year.

I hope you are as delighted as I was when I first learned how far along Apple actually is in removing toxic chemicals from its products and recycling its older products. We apologize for leaving you in the dark for this long. Apple is already a leader in innovation and engineering, and we are applying these same talents to become an environmental leader. Based on our tangible actions and results over time, hopefully our customers, employees, shareholders and professional colleagues will all feel proud of our ongoing efforts to become a greener Apple.