We will use the given sentence to form a conditional, then its converse, inverse, and contrapositive before finally writing a biconditional. To start, we shall identify the hypothesis and conclusion of the sentence.
&Hypothesis: an animal is a monkey
&Conclusion: it is a mammal
Now that we can the hypothesis and conclusion, we can use the following information to form each of the requested statements. Here, p is the original hypothesis and q is the original conclusion.
| Conditional
|
Converse
|
Inverse
|
Contrapositive
|
Biconditional
|
| p→ q
|
q→ p
|
~ p→ ~ q
|
~ q→ ~ p
|
p↔ q
|
With this information, we are now able to plug in the statements that we found above and form each of the requested statements.
The conditional
Let's start by writing the conditional, which takes the form of If-then. Recall that the hypothesis follows the word If and the conclusion follows the word then.
Ifan animal is a monkey,
thenit is a mammal.
Now we can form the converse.
The converse
To write the converse of a conditional statement, interchange the places of the hypothesis and the conclusion.
Ifan animal is a mammal,
thenit is a monkey.
Time to write the inverse!
The inverse
To write the inverse of a conditional statement, we negate both the hypothesis and the conclusion. This means each of the statements should state the opposite. We usually do this by adding or removing a not.
Ifan animal is not a monkey,
thenit is not a mammal.
Let's now focus on the contrapositive.
The contrapositive
To write the contrapositive of a conditional statement, first recall the converse we wrote above:
Ifan animal is a mammal,
thenit is a monkey.
To write the contrapositive, we negate the hypothesis and conclusion of the converse.
Ifan animal is not a mammal,
thenit is not a monkey.
Almost done! It is time to write the biconditional.
The biconditional
The biconditional is a statement that uses if and only if to connect the hypothesis and conclusion.
An animal is a monkey
if and only ifit is a mammal.
