Question

The compound statement, “If you won the race; then you did not run faster than others.” Is equivalent to
A.If you won the race, then you ran faster than others.
B.If you ran faster than others, then you won the race.
C.If you did not win the race, then you did not run faster than others.
D.If you ran faster than others, then you did not win the race.

Answer 1

Hint: We will read the given compound statement. We will identify the component statements of the compound statement. We will use the rules of mathematical reasoning to find its equivalent statement.

Complete step-by-step answer:
We know that there are 2 types of mathematical reasoning – Inductive and deductive. We will use deductive reasoning in this problem. We know that a sentence is a mathematically accepted statement if it is true or false but not both. We know that a compound statement is a statement which is made up of 2 or more statements. The statements that make up a compound statement are called component statements. We know that the compound statements are combined using connecting words like ‘and’/ ‘or’ to form a compound statement.
We can say that in the problem ‘If you won the race’ is p and ‘you did not run faster than others’ is q.
So, according to the rule, the given statement will be equivalent to “If you ran faster than others, then you did not win the race.
The equivalent statement is known as the contrapositive of the original statement.
$\therefore $ Option D is the correct option.

Note: We must be careful that if p implies q, then it is not necessary that (not p) will imply (not q). Similarly, if p implies q, then it is not necessary that q implies p. Here we need to understand the given statement first. In the statement it says one has won the race because they did not run faster. In the statement, the first half says the opposite of the second half. So, we have to search for that statement that has two parts and both the parts are opposite to each other.