Consider the proposition ‘If Ayesha gets 75% or more in the examination, then she will get an A grade for the course.’. We can write this statement as ‘If p, and q’, where
p: Ayesha gets 75% or more in the examination, and
q: Ayesha will get an A grade for the course.
This compound statement is an example of the implication of q by p.
Definition: Given any two propositions p and q, we denote the statement ‘If p, then q’ by p → q. We also read this as ‘p implies q’. or ‘p is sufficient for q’, or ‘p only if q’. We also call p the hypothesis and q the conclusion. Further, a statement of the form p → q is called a conditional statement or a conditional proposition. So, for example, in the conditional proposition ‘If m is in Z, then m belongs to Q.’ the hypothesis is ‘m ∈ Z’ and the conclusion is ‘m ∈ Q’. Mathematically, we can write this statement as
m ∈ Z → m ∈ Q.
Let us analyse the statement p → q for its truth value. Do you agree with the truth table we’ve given below (Table 3)? You may like to check it out while keeping an
example from your surroundings in mind.
You may wonder about the third row in Table 3. But, consider the example ‘3 < 0 →5 > 0’. Here the conclusion is true regardless of what the hypothesis is. And
therefore, the conditional statement remains true. In such a situation we say that the conclusion is vacuously true.Edit