If the stars are not visible, then it is not night.
False
Contrapositive statement
If it is not night then the stars are visible.
True
Practice makes perfect
Let's consider each of the statements one at a time using the given p and q.
p =& the stars are visible
q =& it is night
Conditional Statement
We can write the conditional statement, p→ q, in an if-then form.
If the stars are visible,
then it is night.
When we can see stars, we know it is night. This 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 night,
then the stars are visible.
When it is cloudy or there is lots of light pollution, even though it is night, we cannot necessarily can see the stars. Therefore, this statement is false.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If the stars are not visible,
then it is not night.
Similar to the converse, the stars can be obscured at night due to clouds or light pollution. This is also false.
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.
If it is not night,
then the stars are not visible.
Since the stars are visible only at night, this statement is true.
