Introduction: Logical Reasoning - SS2 Mathematics Lesson Note
Recall that given two statements \(p,q\), the compound statement “if \(p\) then \(q\)” denoted by \(p \Rightarrow q\) is called a conditional statement or implication.
Such statements are common in mathematics and are used in proposition or theorem. Consider the Pythagoras theorem: if \(\mathrm{\Delta}ABC\) is a right angled triangle with sides \(a,b,c\) and \(c\) is the hypotenuse then \(a^{2}{+ b}^{2}{= c}^{2}\).
Again, consider the conditional statements \(p \Rightarrow q\) of \(p,q\), the:
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Converse of \(p \Rightarrow q\) is the conditional statement \(q \Rightarrow p\)
Inverse of \(p \Rightarrow q\) is the conditional statement \(\sim p \Rightarrow \sim q\)
Contrapositive of \(p \Rightarrow q\) is the conditional statement \(\sim q \Rightarrow \sim p\)
Example 1
\[Let\ p:\ 2 + 3 = 5\]
\[Let\ q:\ 7\ is\ an\ odd\ number\]
| \[p \Rightarrow q\] | If \(2 + 3 = 5\) then \(7\ is\ an\ odd\ number\) | |
|---|---|---|
| Converse | \[q \Rightarrow p\] | If \(7\ is\ an\ odd\ number\) then \(2 + 3 = 5\) |
| Inverse | \[\sim p \Rightarrow \sim q\] | If \(2 + 3 \neq 5\) then \(7\ is\ not\ an\ odd\ number\) |
| Contrapositive | \[\sim q \Rightarrow \sim p\] | If \(7\ is\ not\ an\ odd\ number\) then \(2 + 3 \neq 5\) |
The truth table for the above can be represented thus:
| Implication | Converse | Inverse | Contrapositive | ||||
|---|---|---|---|---|---|---|---|
| \[\mathbf{p}\] | \[\mathbf{q}\] | \[\mathbf{\sim p}\] | \[\mathbf{\sim q}\] | \[\mathbf{p \Rightarrow q}\] | \[\mathbf{q}\mathbf{\Rightarrow}\mathbf{p}\] | \[\mathbf{\sim p}\mathbf{\Rightarrow}\mathbf{\sim q}\] | \[\mathbf{\sim q}\mathbf{\Rightarrow \sim p}\] |
| \[T\] | \[T\] | \[F\] | \[F\] | \[T\] | \[T\] | \[T\] | \[T\] |
| \[T\] | \[F\] | \[F\] | \[T\] | \[F\] | \[T\] | \[T\] | \[F\] |
| \[F\] | \[T\] | \[T\] | \[F\] | \[T\] | \[F\] | \[F\] | \[T\] |
| \[F\] | \[F\] | \[T\] | \[T\] | \[T\] | \[T\] | \[T\] | \[T\] |
The truth table shows that \(p \Rightarrow q\) (implication) and \(\sim q \Rightarrow \sim p\) (contrapositive) are equivalent. We can prove that an implicative statement (proposition or theorem) is equivalent to its contrapositive. This method of proof is called indirect proof.
Note also, that the converse and inverse are equivalent to each other.