Consider two propositions generated by*p*and*q:* *¬*(p*∧q)*and*¬p∨ ¬q. At first*
glance, they are different propositions. In form, they are different, but they
have the same meaning. One way to see this is to substitute actual propositions
for*p*and*q; such asp: I’ve been to Toronto; andq: I’ve been to Chicago.*

Then*¬*(p*∧q)*translates to “I haven’t been to both Toronto and Chicago,”
while *¬p∨ ¬q* is “I haven’t been to Toronto or I haven’t been to Chicago.”
Determine the truth values of these propositions. Naturally, they will be true
for some people and false for others. What is important is that no matter
what truth values they have, *¬*(p*∧q)* and *¬p∨ ¬q* will have the same truth
value. The easiest way to see this is by examining the truth tables of these
propositions.
*p* *q* *¬*(p*∧q)* *¬p∨ ¬q*
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0

**Table 3.3.1:** Truth Tables for*¬*(p*∧q)*and*¬p∨ ¬q*

In all four cases, *¬*(p*∧q)*and*¬p∨ ¬q*have the same truth value. Further-
more, when the biconditional operator is applied to them, the result is a value
of true in all cases. A proposition such as this is called a tautology.

**3.3.1 Tautologies and Contradictions**

**Definition 3.3.2 Tautology.** An expression involving logical variables that
is true in all cases is a tautology. The number 1 is used to symbolize a tautology.
♢
**Example 3.3.3** **Some Tautologies.** All of the following are tautologies
because their truth tables consist of a column of 1’s.

(a) (*¬*(p*∧q))↔*(*¬p∨ ¬q).*
(b) *p∨ ¬p*
(c) (p*∧q)→p*
(d) *q→*(p*∨q)*
(e) (p*∨q)↔*(q*∨p)*
□
**Definition 3.3.4 Contradiction.** An expression involving logical variables
that is false for all cases is called a contradiction. The number 0 is used to

CHAPTER 3. LOGIC 48
**Example 3.3.5 Some Contradictions.** *p∧ ¬p*and(p*∨q)∧*(*¬p)∧*(*¬q)*are

contradictions. □

**3.3.2 Equivalence**

**Definition 3.3.6 Equivalence.** Let*S*be a set of propositions and let*r*and
*s*be propositions generated by*S.* *r*and*s* are equivalent if and only if*r↔s*
is a tautology. The equivalence of*r*and*s*is denoted*r* *⇐⇒* *s.* ♢
Equivalence is to logic as equality is to algebra. Just as there are many ways
of writing an algebraic expression, the same logical meaning can be expressed
in many different ways.

**Example 3.3.7 Some Equivalences.** The following are all equivalences:

(a) (p*∧q)∨*(*¬p∧q)* *⇐⇒* *q.*
(b) *p→q* *⇐⇒ ¬q→ ¬p*

(c) *p∨q* *⇐⇒* *q∨p.*

□ All tautologies are equivalent to one another.

**Example 3.3.8 An equivalence to** 1. *p∨ ¬p* *⇐⇒* 1. □
All contradictions are equivalent to one another.

**Example 3.3.9 An equivalence to** 0. *p∧ ¬p* *⇐⇒* 0. □

**3.3.3 Implication**

Consider the two propositions:

*x: The money is behind Door A; and*
*y: The money is behind Door A or Door B.*

**Table 3.3.10**

Imagine that you were told that there is a large sum of money behind one
of two doors marked A and B, and that one of the two propositions*x* and*y*
is true and the other is false. Which door would you choose? All that you
need to realize is that if*x*is true, then*y*will also be true. Since we know that
this can’t be the case,*y*must be the true proposition and the money is behind
Door B.

This is an example of a situation in which the truth of one proposition leads
to the truth of another. Certainly,*y* can be true when*x*is false; but *x*can’t
be true when*y* is false. In this case, we say that*x*implies*y.*

Consider the truth table of*p→q,*Table 3.1.7. If*p*implies*q, then the third*
case can be ruled out, since it is the case that makes a conditional proposition
false.

**Definition 3.3.11 Implication.** Let *S* be a set of propositions and let *r*
and*s* be propositions generated by *S. We say thatr* implies*s* if *r→* *s*is a
tautology. We write*r⇒s*to indicate this implication. ♢
**Example 3.3.12** **Disjunctive Addition.** A commonly used implication
called “disjunctive addition” is *p* *⇒* (p*∨q), which is verified by truth table*

CHAPTER 3. LOGIC 49
Table 3.3.13. □
*p* *q* *p∨q* *p→p∨q*
0 0 0 1
0 1 1 1
1 0 1 1
1 1 1 1

**Table 3.3.13:** Truth Table to verify that*p⇒*(p*∨q)*

If we let*p*represent “The money is behind Door A” and*q* represent “The
money is behind Door B,”*p⇒*(p*∨q)*is a formalized version of the reasoning
used in Example 3.3.12. A common name for this implication is disjunctive
addition. In the next section we will consider some of the most commonly used
implications and equivalences.

When we defined what we mean by a Proposition Generated by a Set, we
didn’t include the conditional and biconditional operators. This was because of
the two equivalences*p→q⇔ ¬p∨q*and*p↔q⇔*(p*∧q)∨*(*¬p∧¬q). Therefore,*
any proposition that includes the conditional or biconditional operators can be
written in an equivalent way using only conjunction, disjunction, and negation.
We could even dispense with disjunction since*p∨q*is equivalent to a proposition
that uses only conjunction and negation.

**3.3.4 A Universal Operation**

We close this section with a final logical operation, the Sheffer Stroke, that has the interesting property that all other logical operations can be created from it. You can explore this operation inExercise 3.3.5.8

**Definition 3.3.14 The Sheffer Stroke.** The Sheffer Stroke is the logical
operator defined by the following truth table:

*p* *q* *p|q*

0 0 1

0 1 1

1 0 1

1 1 0

**Table 3.3.15:** Truth Table for the Sheffer Stroke

♢

**3.3.5 Exercises for Section 3.3**

**1.** Given the following propositions generated by*p,q, andr, which are equiv-*
alent to one another?

(a) (p*∧r)∨q*
(b) *p∨*(r*∨q)*
(c) *r∧p*
(d) *¬r∨p*
(e) (p*∨q)∧*(r*∨q)*
(f) *r→p*
(g) *r∨ ¬p*
(h) *p→r*
**2.**

CHAPTER 3. LOGIC 50
(b) Give an example other than*x*itself of a proposition generated by*p,*

*q, andr* that is equivalent to*x.*

(c) Give an example of a proposition other than*x*that implies*x.*
(d) Give an example of a proposition other than*x*that is implied by*x.*
**3.** Is an implication equivalent to its converse? Verify your answer using a

truth table.

**4.** Suppose that*x*is a proposition generated by*p,q, andr*that is equivalent
to*p∨ ¬q. Write out the truth table forx.*

**5.** How large is the largest set of propositions generated by*p*and*q*with the
property that no two elements are equivalent?

**6.** Find a proposition that is equivalent to*p∨q* and uses only conjunction
and negation.

**7.** Explain why a contradiction implies any proposition and any proposition
implies a tautology.

**8.** The significance of the Sheffer Stroke is that it is a “universal” operation
in that all other logical operations can be built from it.

(a) Prove that *p|q*is equivalent to *¬*(p*∧q).*
(b) Prove that*¬p⇔p|p.*

(c) Build *∧*using only the Sheffer Stroke.
(d) Build*∨*using only the Sheffer Stroke.