If you are not an only child, then you have a sibling.
True
Converse statement
If you have a sibling, then you are not an only child.
True
Inverse statement
If you are an only child, then you do not have a sibling.
True
Contrapositive statement
If you do not have a sibling, then you are an only child.
True
Practice makes perfect
Let's consider each of the statements one at a time using the given p and q.
p =& you are not an only child
q =& you have a sibling
Conditional Statement
We can write the conditional statement, p→ q, in an if-then form.
If you are not an only child,
then you have a sibling.
This is a true statement, as not being an only child necessarily means that you have at least one sibling.
Converse
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
If you have a sibling,
then you are not an only child.
By the same logic that 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. Note that the hypothesis already contains a "not". Therefore, when we negate the hypothesis we get a double negation which is the same thing as erasing the original not.
If you are an only child,
then you do not have a sibling.
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 you do not have a sibling,
then you are an only child.
By the same logic we know that the previous statements are true, we know that the contrapositive is true.
