Question

Determine the contrapositive of each of the following statements:
If he has courage he will win.

Answer 1

Hint: Now we know that the statement is in the form of if p then q.
Now contrapositive of ‘if p then q’ is given by ‘if not q, then not p’ hence using this we will find the contrapositive of the given statement.

Complete step-by-step answer:
Now the given statement is if he has courage he will win.
We can write this statement as if he has courage then he will win
Now let p be the statement he has courage and q be the statement he will win.
Now this statement is in the form of if p then q.
Now we know that contrapositive of the statements ‘if p then q’ is given by ‘if not q, then not p’
Symbolically we have the given statement as $p\Rightarrow q$ read as p implies q.
Now similarly contrapositive is written as $\sim q\Rightarrow \sim p$ read as not p implies not q.
Now we have p be the statement he has courage and q be the statement he will win.
Now $\sim p$ or $\sim q$ means negation of p. hence we will have to add not to the task.
Hence $\sim p$ read as not p is = he doesn’t have courage.
And $\sim q$ read as not q is = he will not win.
Hence $\sim q\Rightarrow \sim p$ He will not win implies he doesn’t have courage
Hence the contrapositive of the statement is he will not win if he doesn’t have courage.

Note: The given statement is if he has courage he will win. It is quite tempting to write the contrapositive as if he has no courage he will not win. But this would be incorrect as the contrapositive of statement ‘if p then q’ is given by ‘if not q, then not p’