b How do you write the converse, the inverse and the contrapositive to a biconditional statement?
A
aConditional Statement: If there is a plane, then it contains at least three noncollinear points
B
b See solution.
Practice makes perfect
a An if-then statement contains a hypothesis and conclusion. The Plane-Point Postulate states that a plane contains at least three noncollinear points. This means if we start with a plane, our hypothesis, we are able to place at least three noncollinear points on it, our conclusion. With this, we can write the postulate in if-then form.
If there is a plane
then it contains at least
three noncollinear points
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 are at least
three noncollinear points,
then we can draw a plane.This is true for the same reason the Plane-Point Postulate is true.
Inverse
The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
If there is not a plane
then it does not contain at
least three noncollinear points
This is also true for the same reason the Plane-Point Postulate is true. We cannot have a plane without at least three noncollinear points.
Contrapositive
The contrapositive of a conditional statement, ~ q→ ~ p, is similar to the converse except we have to negate both the hypothesis and the conclusion.
If there are not at least three
noncollinear points,
then we cannot draw a plane.
This is true for the same reason the inverse is true. We need at least three noncollinear points to have a plane.
