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Contrapositive Proof: A Technique to Prove Conditional Statements, Lecture notes of Logic

Advanced Technology InstituteLogic

The concept of contrapositive proof, a method used to prove conditional statements in mathematics. It discusses the logic behind contrapositive proof, its setup, and provides examples to illustrate its use. The document also contrasts it with direct proof.

Typology: Lecture notes

2021/2022

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CHAPTER 5
Contrapositive Proof
We now examine
an alternative to direct proof called
contrapositive
proof
. Like direct proof, the technique of contrapositive proof is
used to prove conditional statements of the form “If
P
, then
Q
.” Although
it is possible to use direct proof exclusively, there are occasions where
contrapositive proof is much easier.
5.1 Contrapositive Proof
To understand how contrapositive proof works, imagine that you need to
prove a proposition of the following form.
Proposition If P, then Q.
This is a conditional statement of form
P⇒Q
. Our goal is to show
that this conditional statement is true. Recall that in Section 2.6 we
observed that
P⇒Q
is logically equivalent to
∼Q⇒∼P
. For convenience,
we duplicate the truth table that verifies this fact.
P Q ∼Q∼P P ⇒Q∼Q⇒∼ P
T T F F T T
T F T F F F
F T F T T T
F F T T T T
According to the table, statements
P⇒Q
and
∼Q⇒∼P
are different
ways of expressing exactly the same thing. The expression
∼Q⇒∼ P
is
called the contrapositive form of P⇒Q.1
1
Do not confuse the words contrapositive and converse. Recall from Section 2.4 that the
converse of P⇒Qis the statement Q⇒P, which is not logically equivalent to P⇒Q.
pf3
pf4
pf5
pf8
pf9

Partial preview of the text

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CHAPTER 5

Contrapositive Proof

W

e now examine an alternative to direct proof called contrapositive

proof. Like direct proof, the technique of contrapositive proof is

used to prove conditional statements of the form “If P , then Q .” Although

it is possible to use direct proof exclusively, there are occasions where

contrapositive proof is much easier.

5.1 Contrapositive Proof

To understand how contrapositive proof works, imagine that you need to

prove a proposition of the following form.

Proposition If P, then Q.

This is a conditional statement of form P ⇒ Q. Our goal is to show

that this conditional statement is true. Recall that in Section 2.6 we

observed that P ⇒ Q is logically equivalent to ∼ Q ⇒∼ P. For convenience,

we duplicate the truth table that verifies this fact.

P Q ∼ Q ∼ P P ⇒ Q ∼ Q ⇒∼ P

T T F F T T T F T F F F F T F T T T F F T T T T

According to the table, statements P ⇒ Q and ∼ Q ⇒∼ P are different

ways of expressing exactly the same thing. The expression ∼ Q ⇒∼ P is

called the contrapositive form of P ⇒ Q.^1

(^1) Do not confuse the words contrapositive and converse. Recall from Section 2.4 that the converse of P ⇒ Q is the statement Q ⇒ P, which is not logically equivalent to P ⇒ Q.

Contrapositive Proof 103

Since P ⇒ Q is logically equivalent to ∼ Q ⇒∼ P, it follows that to prove

P ⇒ Q is true, it suffices to instead prove that ∼ Q ⇒∼ P is true. If we were

to use direct proof to show ∼ Q ⇒∼ P is true, we would assume ∼ Q is true

use this to deduce that ∼ P is true. This in fact is the basic approach of

contrapositive proof, summarized as follows.

Outline for Contrapositive Proof

Proposition If P, then Q.

Proof. Suppose ∼ Q.

Therefore ∼ P. ■

So the setup for contrapositive proof is very simple. The first line of the

proof is the sentence “Suppose Q is not true.” (Or something to that effect.)

The last line is the sentence “Therefore P is not true.” Between the first

and last line we use logic and definitions to transform the statement ∼ Q

to the statement ∼ P.

To illustrate this new technique, and to contrast it with direct proof,

we now prove a proposition in two ways: first with direct proof and then

with contrapositive proof.

Proposition Suppose x ∈ Z. If 7 x + 9 is even, then x is odd.

Proof. (Direct) Suppose 7 x + 9 is even.

Thus 7 x + 9 = 2 a for some integer a.

Subtracting 6 x + 9 from both sides, we get x = 2 a − 6 x − 9.

Thus x = 2 a − 6 x − 9 = 2 a − 6 x − 10 + 1 = 2(a − 3 x − 5) + 1.

Consequently x = 2 b + 1 , where b = a − 3 x − 5 ∈ Z.

Therefore x is odd. ■

Here is a contrapositive proof of the same statement:

Proposition Suppose x ∈ Z. If 7 x + 9 is even, then x is odd.

Proof. (Contrapositive) Suppose x is not odd.

Thus x is even, so x = 2 a for some integer a.

Then 7 x + 9 = 7(2a) + 9 = 14 a + 8 + 1 = 2(7a + 4) + 1.

Therefore 7 x + 9 = 2 b + 1 , where b is the integer 7 a + 4.

Consequently 7 x + 9 is odd.

Therefore 7 x + 9 is not even. ■

Congruence of Integers 105

Proposition Suppose x, y ∈ Z. If 5 - x y, then 5 - x and 5 - y.

Proof. (Contrapositive) Suppose it is not true that 5 - x and 5 - y.

By DeMorgan’s law, it is not true that 5 - x or it is not true that 5 - y.

Therefore 5 | x or 5 | y. We consider these possibilities separately.

Case 1. Suppose 5 | x. Then x = 5 a for some a ∈ Z.

From this we get x y = 5(a y), and that means 5 | x y.

Case 2. Suppose 5 | y. Then y = 5 a for some a ∈ Z.

From this we get x y = 5(ax), and that means 5 | x y.

The above cases show that 5 | x y, so it is not true that 5 - x y. ■

5.2 Congruence of Integers

This is a good time to introduce a new definition. It is not necessarily

related to contrapositive proof, but introducing it now ensures that we

have a sufficient variety of exercises to practice all our proof techniques on.

This new definition occurs in many branches of mathematics, and it will

surely play a role in some of your later courses. But our primary reason

for introducing it is that it will give us more practice in writing proofs.

Definition 5.1 Given integers a and b and an n ∈ N, we say that a and b

are congruent modulo n if n | (a − b). We express this as a ≡ b (mod n).

If a and b are not congruent modulo n, we write this as a 6 ≡ b (mod n).

Example 5.1 Here are some examples:

1. 9 ≡ 1 (mod 4 ) because 4 | (9 − 1).

2. 6 ≡ 10 (mod 4 ) because 4 | (6 − 10).

3. 14 6 ≡ 8 (mod 4 ) because 4 - (14 − 8).

4. 20 ≡ 4 (mod 8 ) because 8 | (20 − 4).

5. 17 ≡ − 4 (mod 3 ) because 3 | (17 − (−4)).

In practical terms, a ≡ b (mod n) means that a and b have the same

remainder when divided by n. For example, we saw above that 6 ≡ 10

(mod 4) and indeed 6 and 10 both have remainder 2 when divided by 4.

Also we saw 14 6 ≡ 8 (mod 4 ), and sure enough 14 has remainder 2 when

divided by 4 , while 8 has remainder 0.

To see that this is true in general, note that if a and b both have the

same remainder r when divided by n, then it follows that a = kn + r and

b = n + r for some k, ∈ Z. Then a − b = (kn + r) − ( n + r) = n(k − ). But

a − b = n(k − ` ) means n | (a − b), so a ≡ b (mod n). Conversely, one of the

exercises for this chapter asks you to show that if a ≡ b (mod n), then a

and b have the same remainder when divided by n.

106 Contrapositive Proof

We conclude this section with several proofs involving congruence of

integers, but you will also test your skills with other proofs in the exercises.

Proposition Let a, b ∈ Z and n ∈ N. If a ≡ b (mod n), then a^2 ≡ b^2 (mod n).

Proof. We will use direct proof. Suppose a ≡ b (mod n).

By definition of congruence of integers, this means n | (a − b).

Then by definition of divisibility, there is an integer c for which a − b = nc.

Now multiply both sides of this equation by a + b.

a − b = nc (a − b)(a + b) = nc(a + b) a^2 − b^2 = nc(a + b)

Since c(a + b) ∈ Z, the above equation tells us n | (a^2 − b^2 ).

According to Definition 5.1, this gives a^2 ≡ b^2 (mod n). ■

Let’s pause to consider this proposition’s meaning. It says a ≡ b (mod n)

implies a^2 ≡ b^2 (mod n). In other words, it says that if integers a and b

have the same remainder when divided by n, then a^2 and b^2 also have

the same remainder when divided by n. As an example of this, 6 and 10

have the same remainder (2) when divided by n = 4 , and their squares

36 and 100 also have the same remainder (0) when divided by n = 4. The

proposition promises this will happen for all a, b and n. In our examples

we tend to concentrate more on how to prove propositions than on what

the propositions mean. This is reasonable since our main goal is to learn

how to prove statements. But it is helpful to sometimes also think about

the meaning of what we prove.

Proposition Let a, b, c ∈ Z and n ∈ N. If a ≡ b (mod n), then ac ≡ bc (mod n).

Proof. We employ direct proof. Suppose a ≡ b (mod n). By Definition 5.1,

it follows that n | (a − b). Therefore, by definition of divisibility, there exists

an integer k for which a − b = nk. Multiply both sides of this equation

by c to get ac − bc = nkc. Thus ac − bc = n(kc) where kc ∈ Z, which means

n | (ac − bc). By Definition 5.1, we have ac ≡ bc (mod n). ■

Contrapositive proof seems to be the best approach in the next example,

since it will eliminate the symbols - and 6 ≡.

108 Contrapositive Proof

Mathematical statements (equations, etc.) are like English phrases that

happen to contain special symbols, so use normal punctuation.

3. Separate mathematical symbols and expressions with words.

Not doing this can cause confusion by making distinct expressions

appear to merge into one. Compare the clarity of the following examples.

Because x^2 − 1 = 0 , x = 1 or x = − 1. ×

Because x^2 − 1 = 0 , it follows that x = 1 or x = − 1. X

Unlike A ∪ B, A ∩ B equals ;. ×

Unlike A ∪ B, the set A ∩ B equals ;. X

4. Avoid misuse of symbols. Symbols such as =, ≤, ⊆, ∈, etc., are not

words. While it is appropriate to use them in mathematical expressions,

they are out of place in other contexts.

Since the two sets are =, one is a subset of the other. ×

Since the two sets are equal, one is a subset of the other. X

The empty set is a ⊆ of every set. ×

The empty set is a subset of every set. X

Since a is odd and x odd ⇒ x^2 odd, a^2 is odd. ×

Since a is odd and any odd number squared is odd, then a^2 is odd.X

5. Avoid using unnecessary symbols. Mathematics is confusing enough

without them. Don’t muddy the water even more.

No set X has negative cardinality. ×

No set has negative cardinality. X

6. Use the first person plural. In mathematical writing, it is common

to use the words “we” and “us” rather than “I,” “you” or “me.” It is as if

the reader and writer are having a conversation, with the writer guiding

the reader through the details of the proof.

7. Use the active voice. This is just a suggestion, but the active voice

makes your writing more lively.

The value x = 3 is obtained through the division of both sides by 5 .×

Dividing both sides by 5 , we get the value x = 3. X

8. Explain each new symbol. In writing a proof, you must explain the

meaning of every new symbol you introduce. Failure to do this can lead

to ambiguity, misunderstanding and mistakes. For example, consider

the following two possibilities for a sentence in a proof, where a and b

have been introduced on a previous line.

Mathematical Writing 109

Since a | b, it follows that b = ac. ×

Since a | b, it follows that b = ac for some integer c. X

If you use the first form, then a reader who has been carefully following

your proof may momentarily scan backwards looking for where the c

entered into the picture, not realizing at first that it came from the

definition of divides.

9. Watch out for “it.” The pronoun “it” can cause confusion when it is

unclear what it refers to. If there is any possibility of confusion, you

should avoid the word “it.” Here is an example:

Since X ⊆ Y , and 0 < |X |, we see that it is not empty. ×

Is “it” X or Y? Either one would make sense, but which do we mean?

Since X ⊆ Y , and 0 < |X |, we see that Y is not empty. X

10. Since, because, as, for, so. In proofs, it is common to use these

words as conjunctions joining two statements, and meaning that one

statement is true and as a consequence the other true. The following

statements all mean that P is true (or assumed to be true) and as a

consequence Q is true also.

Q since P Q because P Q, as P Q, for P P, so Q

Since P, Q Because P, Q as P, Q

Notice that the meaning of these constructions is different from that of

“If P , then Q ,” for they are asserting not only that P implies Q, but also

that P is true. Exercise care in using them. It must be the case that P

and Q are both statements and that Q really does follow from P.

x ∈ N, so Z ×

x ∈ N, so x ∈ Z X

11. Thus, hence, therefore consequently. These adverbs precede a

statement that follows logically from previous sentences or clauses. Be

sure that a statement follows them.

Therefore 2 k + 1. ×

Therefore a = 2 k + 1. X

12. Clarity is the gold standard of mathematical writing. If you

believe breaking a rule makes your writing clearer, then break the rule.

Your mathematical writing will evolve with practice useage. One of the

best ways to develop a good mathematical writing style is to read other

people’s proofs. Adopt what works and avoid what doesn’t.