A statement is a good definition, if it is reversible.
D
Practice makes perfect
A statement is a good definition, if it is reversible. In order to decide which statement is a good definition, we will write it as a conditional statement.
If $ p,$ then $ q.$
After that, we will write the converse of the conditional statement.
If $ q,$ then $ p.$
If the conditional statement and its converse are true, we can say that the statement is reversible and it is a good definition. Let's examine each statement and give counterexamples if the converse of the statement is incorrect. The first statement is given as below.
A : Rectangles are usually longer than they are wide.
Let's write it as a conditional statement and converse of the conditional statement.
Conditional Statement: If figures are rectangles, then they are longer than they are wide.
Converse: If figures are longer than they are wide, then they are rectangles.
The converse of the conditional statement is false so it is not reversible. We can give a counterexample for it as the following.
Counterexample: A trapezoid can also be longer than it is wide.
As a result, Statement A is not a good definition. We can examine rest of the statements in the same way to decide which one is a good definition.
Statement
Conditional
Converse
Counterexample
Reversible
A
If figures are rectangles, then they are longer than they are wide.
If figures are longer than they are wide, then they are rectangles.
A trapezoid can also be longer than it is wide.
NO
B
If figures are squares, then they are convex.
If figures are convex, then they are squares.
A rectangle is also convex.
NO
C
If figures are circles, then they have no corners.
If figures have no corners, then they are circles.
An ellipse also has no corners.
NO
D
If polygons are triangles, then they have three sides.
If polygons have three sides, then they are triangle.
