Daytime is defined as the time between sunrise and sunset.
Type of statement
Statement
True or False?
Conditional statement
If the Sun is out, then it is daytime.
True
Converse statement
If it is daytime, then the Sun is out.
True
Inverse statement
If the Sun is not out, then it is not daytime.
True
Contrapositive statement
If it is not daytime, then the Sun is not out.
True
Practice makes perfect
Let's consider each of the statements one at a time using the given p and q.
p =& The Sun is out
q =& It is daytime
Conditional Statement
We can write the conditional statement, p→ q, in an if-then form.
If the Sun is out,
then it is daytime.
Daytime is defined as the time between sunrise and sunset. Therefore, the conditional is a true statement.
Converse
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
If it is daytime,
then the Sun is out.
By the same logic the conditional statement is true, the converse of the conditional statement is also true.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If the Sun is not out,
then it is not daytime.
By the same logic we know that the conditional and converse statements are true, we also know that the inverse is true.
Contrapositive
The contrapositive of a conditional statement, ~ q→ ~ p, starts out with the converse of the conditional statement. Then we have to negate the hypothesis and the conclusion. Again, notice that negating the original hypothesis results in a double negation.
If it is not daytime,
then the Sun is not out.
By the same logic we know that the previous statements are true, we know that the contrapositive is true.
