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.
Table 3: Truth
table for implication
P

Q

p → q

T

T

T

T

F

F

F

T

T

F

F

T

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.
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