Inverse, Converse, Contrapositive

Conditional Statements (Lesson 2-2 textbook)

Conditionals (if-then) and Related Conditionals

A conditional statement is a statement that can be written in "if-then" form.

In general, a conditional statement uses the form "If p, then q".

• The first part of the conditional statement is the hypothesis. The hypothesis follows the word "if".

• The second part of the conditional statement is the conclusion. The conclusion follows the word "then".

Watch the YouTube video "Geometry Conditional Statements" by Christine Hinkley. Here's the link: https://youtu.be/zTHnMTzPEoE

• Vocabulary: conditional statement, hypothesis, conclusion, negation, converse, inverse, contrapositive, biconditional, logically equivalent.

• To write the converse of a conditional statement, simply switch the hypothesis and conclusion.

• To write the inverse of a conditional statement, negate (change to the opposite) both the hypothesis and the conclusion.

• To write the contrapositive of a conditional statement, switch and negate the hypothesis and conclusion. The contrapositive is a combination of the converse and inverse.

Geometry textbook, p.122 Related Conditionals table

Geometry textbook, p.123, logically equivalent statements

p.123

Notice from the previous slides:

• A conditional statement and its contrapositive are always logically equivalent (have the same truth values).

• The converse and inverse of a conditional statement are always logically equivalent.

Conditional Statement: If p, then q.

If you live in West Virginia, then you live in the USA.

• Hypothesis: "you live in West Virginia"

• Conclusion: "you live in the USA"

• Notice that the hypothesis does NOT include the word "if".

• Notice that the conclusion does NOT include the word "then".

Conditional Statement: If p, then q.

If you live in WV, then you live in the USA.

• Converse (if q, then p): If you live in the USA, then you live in WV.

• Inverse (if not p, then not q): If you do not live in WV, then you do not live in the USA.

• Contrapositive (if not q, then not p): If you do not live in the USA, then you do not live in WV.

• Notice that the conditional statement and its contrapositive are both true statements, while the converse and inverse are both false statements (e.g., someone who lives in New York).

Multiple Choice

The negation of the statement "A week has 7 days" is:

(only 1 correct choice)

Hint: The negation of a statement is the opposite of the statement.

• A week has 7 days

• A week does not have 7 days

• A week has fewer than 7 days

• A week has more than 7 days

Multiple Choice

Identify the hypothesis of the conditional statement:

If an angle is right, then its measure is  $$90\degree$$ .

Remember that the hypothesis does not include the word "if".

• an angle is right

• its measure is  $$90\degree$$

• if an angle is right

• then its measure is  $$90\degree$$

Multiple Choice

Identify the conclusion of the conditional statement:  

If an angle is right, then its measure is  $$90\degree$$ .

Remember that the conclusion does not include the word "then".

• an angle is right

• its measure is  $$90\degree$$

• if an angle is right

• then its measure is  $$90\degree$$

Multiple Choice

Identify the converse of the conditional statement:

If an angle is right, then its measure is  $$90\degree$$ .

Hint: The converse of "If p, then q" is "If q, then p".

• If an angle is right, then its measure is not  $$90\degree$$ . 

• If an angle measures  $$90\degree$$ , then it is right.

• If an angle is not right, then its measure is not $$90\degree$$ . 

• If an angle does not measure  $$90\degree$$ , then it is not right.

Multiple Choice

Identify the inverse of the conditional statement:

If an angle is right, then it measures  $$90\degree$$ .

Hint: The inverse of "if p, then q" is "if not p, then not q".

• If an angle is right, then it does not measure  $$90\degree$$ .

• If an angle measures  $$90\degree$$ , then it is right.

• If an angle is not right, then it does not measure  $$90\degree$$ .

• If an angle does not measure  $$90\degree$$ , then it is not right.

Multiple Choice

Identify the contrapositive of the conditional statement:

If an angle is right, then it measures  $$90\degree$$ .

Hint: The contrapositive of "if p, then q" is "if not q, then not p".

• If an angle is right, then it does not measure  $$90\degree$$ .

• If an angle measures  $$90\degree$$ , then it is right.

• If an angle is not right, then it does not measure  $$90\degree$$ .

• If an angle does not measure  $$90\degree$$ , then it is not right.

Multiple Choice

Consider the conditional statement "If today is Saturday, then it's the weekend". 

This is a true statement because the weekend refers to Saturday and Sunday.

Identify the converse statement and its truth value.

[Hint:  The converse of "If p, then q" is "If q, then p".]

• The converse is "If today is not Saturday, then it's not the weekend" and is true.

• The converse is "If today is not Saturday, then it's not the weekend" and is false because it could be Sunday.

• The converse is "If it's the weekend, then today is Saturday" and is false because it could be Sunday.

• The converse is "If it's not the weekend, then today is not Saturday and true.