3. Postulates and Diagrams
Sign In
If there are two points then there exists exactly one line that passes through them
This is false as we can draw an arbitrary line without being given points.
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.
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.