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Section 2.2 Conditionals, Converse, Inverse, and Contrapositive

Many statements and theorems in mathematics are of the form “If \(X\) is true, then \(Y\) is true.”, called a conditional.

Conditionals are the basis for logical deduction, because we want to be able to make a conclusion based on known facts.

Subsection 2.2.1 Conditional (Implication)

Definition 2.2.1.

The conditional (or implication), denoted by \(P \rightarrow Q\text{,}\) is a statement that is true if \(P\) if false or if \(P\) and \(Q\) are true, and is false if \(P\) is true and \(Q\) is false.

In mathematics, \(P\) is often called the hypothesis and \(Q\) is called the conclusion. In logic, \(P\) and \(Q\) are more typically called the antecedent and consequent, respectively.

It is called a conditional because the truth of the conclusion \(Q\) is conditional on the truth of the hypothesis \(P\text{.}\) It is called an implication because the truth of \(P\) implies the truth of \(Q\text{.}\)

“If what you've told me is true, [then] you will have gained my trust.” (Mace Windu).

If a conditional is true, then the hypothesis \(P\) gives a condition (not necessarily the only condition) under which the conclusion \(Q\) will also be guaranteed to be true.

If the condition \(P\) is not met (i.e. is false), a conditional says nothing about the truth of the conclusion.

The truth table for a conditional is given by,

\begin{equation*} \begin{array}{c|c|c} P \amp Q \amp P \rightarrow Q \\ \hline T \amp T \amp T \\ \hline T \amp F \amp F \\ \hline F \amp T \amp T \\ \hline F \amp F \amp T \end{array} \end{equation*}

If a conditional statement is true simply because its hypothesis is false, it is said to be vacuously true (or, true by default).

Alternative forms of the conditional include, “If \(P\text{,}\) then \(Q\)”, “\(P\) implies \(Q\)”, “\(P\) is a sufficient condition for \(Q\)”, “\(P\) only if \(Q\)”, “\(Q\) if \(P\)”, and, “\(Q\) is a necessary condition for \(P\)”.

Subsection 2.2.2 Logical Connectives as Operators

These logical connectives can be thought of as operators that combine multiple statments and ouput another single statement. This is analogous to how \(+\) or \(\times\) combines two numbers and outputs a single number.

Subsection 2.2.3 Converse

Definition 2.2.3.

The converse, \(Q \rightarrow P\text{,}\) switches the hypothesis and conclusion.

\begin{equation*} \begin{array}{c|c|c|c} P \amp Q \amp P \rightarrow Q \amp Q \rightarrow P \\ \hline T \amp T \amp T \amp T \\ \hline T \amp F \amp F \amp T \\ \hline F \amp T \amp T \amp F \\ \hline F \amp F \amp T \amp T \end{array} \end{equation*}

Definition 2.2.4.

The inverse, \(\neg P \rightarrow \neg Q\text{,}\) negates the hypothesis and conclusion.

The truth table is given by,

\begin{equation*} \begin{array}{c|c|c|c|c|c} P \amp Q \amp \neg P \amp \neg Q \amp P \rightarrow Q \amp \neg P \rightarrow \neg Q \\ \hline T \amp T \amp F \amp F \amp T \amp T \\ \hline T \amp F \amp F \amp T \amp F \amp T \\ \hline F \amp T \amp T \amp F \amp T \amp F \\ \hline F \amp F \amp T \amp T \amp T \amp T \end{array} \end{equation*}

Subsection 2.2.4 Contrapositive

Definition 2.2.5.

The contrapositive, \(\neg Q \rightarrow \neg P\text{,}\) negates both statements, and reverses the order of the conditional.

In short, the contrapositive is the converse of the inverse. It has truth table given by,

\begin{equation*} \begin{array}{c|c|c|c|c|c} P \amp Q \amp \neg P \amp \neg Q \amp P \rightarrow Q \amp \neg Q \rightarrow \neg P \\ \hline T \amp T \amp F \amp F \amp T \amp T \\ \hline T \amp F \amp F \amp T \amp F \amp F \\ \hline F \amp T \amp T \amp F \amp T \amp T \\ \hline F \amp F \amp T \amp T \amp T \amp T \end{array} \end{equation*}

Notice that the truth values for the contrapositive \(\neg Q \rightarrow \neg P\) are the same as the original condition \(P \rightarrow Q\text{.}\)