a What do you know about natural arches in other parts of the world?
B
b What kind of statement is always logically equivalent to a conditional statement?
C
c Find the inverse and converse of the conditional statement.
A
aExample Conditional Statement 1: If the largest natural arch is in the United States, then it is in Thompson Utah.
Example Conditional Statement 2: If the largest natural arch is Landscape Arch, then the largest arch spans 290 feet.
B
bContrapositive: If the largest arch does not span 290 feet, then the largest arch is not Landscape Arch.
C
cConverse: If the largest arch spans 290 feet, then the largest arch is Landscape Arch. This is not a true statement, see solution.
Inverse: If the largest arch does not span 290 feet, then the largest arch is not Landscape Arch. This is a true statement, see solution.
Practice makes perfect
a The question does not state anything about whether or not Landscape Arch is the largest natural arch just in the United States or in the entire world. With this information we can write two if-then statements.
if the largest natural arch is
in the United States,
then it is in Thompson Utah.
This is a true conditional statement, because we know from the exercise that the largest natural arch in the United States is in Thompson, Utah.
To write a second if-then statement we can use the fact that Landscape Arch spans 290 feet.
if the largest natural arch is Landscape Arch,
then the largest arch spans 290 feet.
This is also a true conditional statement, because we know from the exercise that Landscape Arch spans 290 feet.
b Note that a conditional statement and its contrapositive are logically equivalent. If the conditional statement is true, so is the contrapositive. The contrapositive of a conditional statement, ~ q→ ~ p, starts out with the converse of the conditional statement.
Converse
If the largest arch spans 290 feet,
then the largest arch is Landscape Arch.
To obtain the contrapositive, we have to negate the hypothesis and the conclusion of the converse.
Contrapositive
If the largest arch does not span 290 feet,
then the largest arch is not Landscape Arch.
c The other two related conditional statements are the converse and inverse of the conditional statement.
Converse
If the largest arch spans 290 feet,
then the largest arch is Landscape Arch.
This is not a true statement, as there could be other arches in the world that span 290 feet. The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement.
Inverse
if the largest natural arch is not
Landscape Arch,
then the largest arch does not
span 290 feet.
This is actually a true statement. If Landscape Arch is not the largest natural arch, then there must be one that is larger than 290 feet.
