Note that "not two points" can be interpreted as less than two points or more than two points.
A
a
Conditional statement: If there are two points, then there exists exactly one line that passes through them
B
b
See solution.
Practice makes perfect
a
An if-then statement contains a hypothesis and conclusion. The Two Point Postulate states that through any two points, there exists exactly one line. This means if we start with two points, our hypothesis, then we are able to draw only one line through them, our conclusion. With this, we can write the postulate in if-then form.
If there are two points
then there exists exactly one
line that passes through them
b
Let's go through the conditional statements one at a time.
Converse
The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement.
If there exists exactly one line
that passes through a given point or points,
then there are two points.This is false as we can draw an arbitrary line without being given points.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If there are not two points
then there is not exactly one
line that passes through them
If we do not have two points, then we must have no points or one point. Since lines are made of points, if there are no points, we do not have a line. If we have only one point, then infinitely many lines can pass through that single point. Thus, this conditional is false.
Contrapositive
The contrapositive of a conditional statement, ~ q→ ~ p, is similar to the converse of the conditional statement except we have to negate both the hypothesis and the conclusion.
If there is not exactly one line
that passes through a given point or points,
then there are not two points.
What this tells us is that we have multiple lines passing through a given point or points. Since we cannot draw multiple lines passing through two points as stated by the Two-Point Postulate, we know that this conditional is true.
