Conditional Statement
How to find the hypothesis, the conclusion, and the truth value of a conditional statement: definition, truth value, 6 examples, and their solutions.
Definition
[p → q] is a conditional statement.
It means [If p, then q].
p is the hypothesis.
q is the conclusion.
Example
The structure of the given statement is
[if ... , then ...].
It's a conditional statement.
So the statement behind if is p:
2 is a prime number.
And the statement behind then is q:
2 is an odd number.
So the hypothesis, p, is
2 is a prime number.
And the conclusion, q, is
2 is an odd number.
So this is the answer.
Example
The structure of the given statement is
[if ... , then ...].
It's a conditional statement.
So the statement behind if is p:
he is not in his room.
And the statement behind then is q:
he is playing basketball.
So the hypothesis, p, is
he is not in his room.
And the conclusion, q, is
he is playing basketball.
So this is the answer.
Example
The structure of the given statement is
[... if ...].
[If] is in the middle of the statement.
But this is still a conditional statement.
So the statement behind if is p:
it's raining.
And q is the former statement:
I'll tell you.
So the hypothesis, p, is
it's raining.
And the conclusion, q, is
I'll tell you.
So this is the answer.
Thuth Value
A conditional statement is false
if p is true and q is false.
(True hypothesis and false conclusion
makes a conditional false.)
Otherwise,
a conditional statment is true.
If the hypothesis, p, is false,
then p → q is true.
It doesn't matter
whether the conclusion q is true or false.
Example
p: 2 is a prime number.
This is true.
q: 2 is a positive number.
This is also true.
Both p and q are true.
So p → q is true.
So true is the answer.
Example
p: 2 is a prime number.
This is true.
r: 2 is an odd number.
This is false.
p is true.
r is false.
So p → r is false.
So false is the answer.
Example
r: 2 is an odd number.
This is false.
The hypothesis r is false.
Then r → p is true.
It doesn't matter
whether the conclusion p is true or not.
So true is the answer.