Question

All Clemson fans root for the Tigers. Rewrite the above statement to reflect the following conditions and state whether each condition is true or false. If false provide a counterexample:
a.) Conditional -
b.) Converse -
c.) Inverse -
d.) Contrapositive -

Answer 1

Given Statement: "All Clemson fans root for the Tigers."

Explanations:

Conditional Statements are statements which hasve two parts in them, namely, the hypothesis part (which comes first) and the conclusion part (which follows the hypothesis). In other words, conditional statements are "if-then" statements.

In this case, "All Clemson fans root for the Tigers." may be written as:

hypothesis part ("if" part) - If a fan is from Clemson

conclusion part ("then" part) - then he/she/they will root for the Tigers.

So the Conditional Statement is : "If a fan is from Clemson, then he/she/they will root for the Tigers."

Converse Statements are statements where the aforementioned hypothesis and conclusion parts are exchanged. That is, hypothesis becomes the conclusion and conclusion becomes the hypothesis.

In this case, the hypothesis part then becomes (the previous conclusion part) - If a fan roots for the Tigers

and the conclusion part (previous hypothesis part) becomes - then he/she/they are from Clemson.

So the Converse Statement is : "If a fan roots for the Tigers, then he/she/they are from Clemson."

Inverse Statements are statments where the conditional statement is negated. This means, we must simply add a "not" to both the hypothesis and the conclusion.

So the Inverse Statement is: "If a fan is not from Clemson, then he/she/they will not root for the Tigers."

Contrapositive Statements are statements where the converse statement is negated. This means, we simply add a "not" to the hypothesis and conclusion of the Converse Statement.

So the Contrapositive Statement is: "If a fan does not root for the Tigers, then he/she/they are not from Clemson."

Now, we analyse which statements are true and false. The given statement implies that if a fan is from Clemson, they support the Tigers. But carefully note that the statement does not say that all Tigers supporters are from Clemson! Indeed, they can be from anywhere.

Understanding this, clearly the conditional statement is true.

But both the Converse and Inverse statements are false! The Converse says that every Tigers fan must indeed be from Clemson which, as we have analysed, is not necessarily true. They can be from anywhere. On the other hand the Inverse says that if a fan is not from Clemson (and say they are from "ABC") then they will necessarily NOT support the Tigers. Which is also false, as Tigers fans may likely be from many places including Clemson, ABC etc).

Two counterexamples can be : ""If a fan roots for the Tigers, then he/she/they can (or may) be from Clemson." and "If a fan is not from Clemson, then he/she/they may not root for the Tigers."

On the other hand, thecontrapositive statement implies that if a fan does NOT support the Tigers, then they can not be from Clemson. Which is absolutely correct as ALL Clemson fans will indeed be Tigers supporters.