The converse, contrapositive and biconditional

Section 2.6 The converse, contrapositive and biconditional

We really want to get to our first proofs, but we need to do a tiny bit more logic, and define a few terms, before we get there. Consider the following three statements derived from implication \(P \implies Q\text{.}\)

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Definition 2.6.1. Contrapositive, converse and inverse.

Let \(P\) and \(Q\) be statements, then:

the statement \((\neg Q) \implies (\neg P)\) is the contrapositive,

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the statement \(Q \implies P\) is the converse, and

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the statement \((\neg P) \implies (\neg Q)\) is the inverse

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of the implication \(P \implies Q\text{.}\)

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The contrapositive and converse appear quite frequently in mathematical writing, but the inverse is rare (in this author’s experience at least). The truth-tables of the implication, contrapositive, converse and inverse are:

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\(P\)	\(Q\)	\(P \implies Q\)	\(\neg Q \implies \neg P\)	\(Q \implies P\)	\(\neg P \implies \neg Q\)
T	T	T	T	T	T
T	F	F	F	T	T
F	T	T	T	F	F
F	F	T	T	T	T

The above tables show that the original implication and the contrapositive have the exactly same truth tables, and that the converse and inverse have the same tables. However we also see that the original implication does not have the same table as the converse or inverse. The inverse is not very commonly used, however the contrapositive and converse will be very useful for us as we continue.

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Remark 2.6.2. Contraposition, conversion and inversion..

Note that the act of forming the contrapositive of \(P\implies Q\) is contraposition. While forming the converse is (sometimes) called conversion, and forming the inverse is called inversion. Notice that the inversion is conversion of the contraposition of the implication.

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Aside: Doing it twice.

While the converse is useful for forming mathematical statements, it can also be the source of bad logic (this is a good moment to go back and look at the warnings Warning 2.5.3 and Warning 2.5.4). The statement

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If he is Shakespeare then he is dead.
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and its converse

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If he is dead then he is Shakespeare.
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definitely do not mean the same thing

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Not every dead person is Shakespeare — ask any Elvis fan.

. However, the converse is often a source of interesting mathematics; once we have proved an implication, we should consider whether or not the converse is true. For example, we have already seen that

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If \(n\) is even then \(n^2\) is even
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is true. It’s converse is

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If \(n^2\) is even then \(n\) is even
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is also true and will turn out to be quite useful later in the text.

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The contrapositive can be extremely helpful — it might be hard to prove the original implication, but much easier to prove the contrapositive. Consider the statement

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If \(n^2\) is odd then \(n\) is odd
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This, it turns out, is awkward to prove as it is stated. Its contrapositive, however, is

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If \(n\) is not odd then \(n^2\) is not odd.
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or equivalently (assuming we are only talking about integers

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Such assumptions happen quite frequently and the reader is often left to infer things from context. Writers do this, not just to be lazy, but so that the text flows and that one is not stating every single assumption in every single statement. That can make reading tedious, toilsome and tiring. Who doesn’t like an alliteration.

)

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If \(n\) is even then \(n^2\) is even
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which, even though we haven’t written up the proof formally, we know is true. Since the truth-table of the contrapositive is identical to the original implication, we now know that

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If \(n^2\) is odd then \(n\) is odd
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must also be true.

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Sometimes both an implication and its converse are true. That is

\begin{gather*} (P \implies Q)\land (Q\implies P) \end{gather*}

This really means that whenever \(P\) is true, so is \(Q\text{,}\) and whenever \(Q\) is true so is \(P\text{.}\) It tells us that there is some sort of equivalence between what is expressed by \(P\) and \(Q\text{.}\) We can rewrite the above statement using the symbol \(\iff\text{.}\) It is our last connective and is called the “biconditional”.

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Definition 2.6.3. The biconditional.

Let \(P\) and \(Q\) be statements. The biconditional, \(P \iff Q\text{,}\) read as “\(P\) if and only if \(Q\)”, is true when \(P\) and \(Q\) have the same truth value and false when \(P\) and \(Q\) take different truth values.

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The biconditional \(P \iff Q\) is also sometimes written as

\(P\) iff \(Q\text{,}\)

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\(P\) is equivalent to \(Q\text{,}\) or

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\(P\) is a necessary and sufficient condition for \(Q\text{.}\)

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Note that “iff” is still read as “if and only if” (and not as “ifffffffffff” with a long “f”-noise).

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Remark 2.6.4.

We noted in the definition above, that \(P \iff Q\) is true when \(P,Q\) have the same truth-values and false when \(P,Q\) have different truth-values. This in turn means that \(P \iff Q\) has the same truth table as the statement \((P \implies Q) \land (Q\implies P)\text{.}\)

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\(P\)	\(Q\)	\(P \iff Q\)	\(P\implies Q\)	\(Q\implies P\)	\((P\implies Q) \land (Q\implies P)\)
T	T	T	T	T	T
T	F	F	F	T	F
F	T	F	T	F	F
F	F	T	T	T	T

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One of the first biconditional statements that we’ll prove is

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\(n^2\) is odd if and only if \(n\) is odd.
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but we have to walk before we run, so armed with all this logic this lets get on to our first proofs. After that we’ll come back to logic.

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