(i)A 0 = A (A A ) = (A A) A = A A = 0.

(iii)A (A B) = (A 0) (A B) = A (0 B) = A 0 = A.

(v)A = A 1 = A (A A ) = (A A) (A A ) = (A A) 0 = A A.

Relations (vii) follows from Proposition 2.12 if we can show that A B is a complement of A B [then it is the complement (A B) ]. Now using part

(i) of this proposition,

(A B) (A B ) = [(A B) A ] [(A B) B ]

=[(A A ) B] [A (B B )]

=[0 B] [A 0]

=0 0

=0.

Similarly, part (ii) gives

(A B) (A B ) = [A (A B )] [B (A B )]

=[(A A ) B ] [(B B ) A ]

=[1 B ] [1 A ]

=1 1

=1.

To prove relation (ix), by definition we have A A = 0 and A A = 1.

PROPOSITIONAL LOGIC

We now show briefly how boolean algebra can be applied to the logic of propositions. Consider two sentences “A” and “B”, which may either be true or false. For example, “A” could be “This apple is red,” and “B” could be “This pear is green.” We can combine these to form other sentences, such as “A and B,” which would be “This apple is red, and this pear is green.” We could also form the sentence “not A,” which would be “This apple is not red.” Let us now compare the truth or falsity of the derived sentences with the truth or falsity of the original ones. We illustrate the relationship by means of a diagram called a truth table. Table 2.4 shows the truth tables for the expressions “A and B,” “A or B,” and “not A.” In these tables, T stands for “true” and F stands for “false.” For

example, if the statement “A” is true while “B” is false, the statement “A and B” will be false, and the statement “A or B” will be true.

We can have two seemingly different sentences with the same meaning; for example, “This apple is not red or this pear is not green” has the same meaning as “It is not true that this apple is red and that this pear is green.” If two sentences, P and Q, have the same meaning, we say that P and Q are logically equivalent, and we write P = Q. The example above concerning apples and pears implies that

(not A) or (not B) = not (A and B).

This equation corresponds to De Morgan’s law in a boolean algebra.

It appears that a set of sentences behaves like a boolean algebra. To be more precise, let us consider a set of sentences that are closed under the operations of “and,” “or,” and “not.” Let K be the set, each element of which consists of all the sentences that are logically equivalent to a particular sentence. Then it can be verified that (K, and, or, not) is indeed a boolean algebra. The zero element is called a contradiction, that is, a statement that is always false, such as “This apple is red and this apple is not red.” The unit element is called a tautology, that is, a statement that is always true, such as “This apple is red or this apple is not red.” This allows us to manipulate logical propositions using formulas derived from the axioms of a boolean algebra.

An important method of combining two statements, A and B, in a sentence is by a conditional, such as “If A, then B,” or equivalently, “A implies B,” which we shall write as “A B.” How does the truth or falsity of such a conditional depend on that of A and B? Consider the following sentences:

1.If x > 4, then x2 > 16.

2.If x > 4, then x2 = 2.

3.If 2 = 3, then 0.2 = 0.3.

4.If 2 = 3, then the moon is made of green cheese.

Clearly, if A is true, then B must also be true for the sentence “A B” to be true. However, if A is not true, then the sentence “If A, then B” has no standard meaning in everyday language. Let us take “A B” to mean that we cannot have A true and B not true. This implies that the truth value of the

statement “A B” is the same as that of “not (A and not B).” Let us write, , and for “and,” “or,” and “not,” respectively. Then “A B” is equivalent to (A B ) = A B. Thus “A B” is true if A is false or if B is true. Using this definition, statements 1, 3, and 4 are all true, whereas statement 2 is false.

We can combine two conditional statements to form a biconditional statement of the form “A if and only if B” or “A B.” This has the same truth value as “(A B) and (B A)” or, equivalently, (A B) (A B ). Another way of expressing this biconditional is to say that “A is a necessary and sufficient condition for B.” It is seen from Table 2.5 that the statement “A B” is true if either A and B are both true or A and B are both false.

Example 2.14. Apply this propositional calculus to determine whether a certain politician’s arguments are consistent. In one speech he states that if taxes are raised, the rate of inflation will drop if and only if the value of the dollar does not fall. On television, he says that if the rate of inflation decreases or the value of the dollar does not fall, taxes will not be raised. In a speech abroad, he states that either taxes must be raised or the value of the dollar will fall and the rate of inflation will decrease. His conclusion is that taxes will be raised, but the rate of inflation will decrease, and the value of the dollar will not fall.

Solution. We write

A to mean “Taxes will be raised,”

B to mean “The rate of inflation will decrease,”

C to mean “The value of the dollar will not fall.”

The politician’s three statements can be written symbolically as

(i)A (B C).

(ii)(B C) A .

(iii)A (C B).

His conclusion is (iv) A B C.

The truth values of the first two statements are equivalent to those of the following:

(i)A ((B C) (B C )).

(ii)(B C) A .

It follows from Table 2.6 that (i) (ii) (iii) (iv) is not a tautology; that is, it is not always true. Therefore, the politician’s arguments are incorrect. They break down when A and C are false and B is true, and when B and C are false and A is true.

SWITCHING CIRCUITS

In this section we use boolean algebra to analyze some simple switching circuits. A switch is a device with two states; state 1 is the “on” state, and state 0 the “off” state. An ordinary household light switch is such a device, but the theory holds equally well for more sophisticated electronic or magnetic two-state devices. We analyze circuits with two terminals: The circuit is said to be closed if current can pass between the terminals, and open if current cannot pass.

We denote a switch A by the symbol in Figure 2.8. We assign the value 1 to A if the switch A is closed and the value 0 if it is open. We denote two switches by the same letter if they open and close simultaneously. If B is a switch that is always in the opposite position to A (that is, if B is open when A is closed, and B is closed when A is open), denote switch B by A .

The two switches A and B in Figure 2.9 are said to be connected in series. If we connect this circuit to a power source and a light as in Figure 2.10, we see that the light will be on if and only if A and B are both switched on; we denote this series circuit by A B. Its effect is shown in Table 2.7.

The switches A and B in Figure 2.11 are said to be in parallel, and this circuit is denoted by A B because the circuit is closed if either A or B is switched on.