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2. Language of Logic
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Geometry

2.02 Converse, inverse, and contrapositive

Lesson
Practice

Converse, inverse, and contrapositive

Conjecture

An unproven statement that is believed to be true but has not yet been proven. A conjecture may be true or false until proven.

A conjecture is a statement that may be true or false until proven. There are many ways to relate, combine, and change conjectures that may change the truth value of statements.

Inverse

A statement formed by negating the hypothesis and conclusion of a conditional statement

Consider the following conditional statement and its inverse:

Statement (If p then q): If an animal is a dog, then it has four legs.

Inverse (If not p then not q): If an animal is not a dog, then it does not have four legs.

Symbolic: If \sim p then \sim q

Converse

A statement formed by switching the hypothesis and conclusion of a conditional statement

Consider the following conditional statement and its converse:

Statement (If p then q): If an animal is a dog, then it has four legs.

Converse (If q then p): If an animal has four legs, then it is a dog.

Symbolic: If q then p

The contrapositive of the statement "If an animal is a dog, then it has four legs" is "If an animal does not have four legs, then it is not a dog." Note that a contrapositive statement is logically equivalent to the original conditional statement.

Contrapositive

A statement formed by interchanging and negating the hypothesis and conclusion of a conditional statement

Consider the following conditional statement and its contrapositive:

Statement (If p then q): If an animal is a dog, then it has four legs.

Contrapositive (If not q then not p): If an animal does not have four legs, then it is not a dog.

Symbolic: If \sim q then \sim p

Examples

Example 1

Consider the conditional statements.

a

"If a number ends in zero, then it is divisible by ten."

State the contrapositive of the statement.

Worked Solution
Create a strategy

The contrapositive of a statement negates both parts of a conditional statement and also switches the order.

Apply the idea

The contrapositive of "If a number ends in zero, then it is divisible by ten" is "If a number is not divisible by ten, it does not end in zero."

b

"If c then d."

Where c and d are both conjectures.

State the converse of the statement.

Worked Solution
Create a strategy

The converse of the conditional statement is formed by switching the hypothesis and conclusion.

Apply the idea

The converse of "If c then d" is "If d then c."

c

"If b then d."

b and d are both conjectures.

State the inverse of the statement.

Worked Solution
Create a strategy

The inverse of the conditional statement is formed by negating the hypothesis and conclusion.

Apply the idea

The inverse of "If b then d" is "If \sim b then \sim d."

Reflect and check

The inverse statement is read as "if not b, then not d".

Idea summary

  • The converse of "if p then q" is "if q then p" and is logically independent of the original

  • The inverse of "if p then q" is "if \sim p then \sim q" and is logically independent of the original

  • The contrapositive of "if p then q" is "if \sim q then \sim p" and is logically equivalent to the original

Law of contrapositive and counterexamples

Exploration

  • Truth is a property of statements (premises and conclusions). Truth is the complete accuracy of each individual statement.

  • Validity requires logical consistency between statements, but it does not require true statements. Validity is a property of the conclusion itself.

Determine whether each statement is true and whether the conclusion can be valid.

  1. Statement 1: If I skip breakfast, then I am hungry.

    Statement 2: If I am hungry, I want pizza.

    Conclusion: Therefore, if I skip breakfast, I want pizza.\,\\\,

  2. Statement 1: Only animals live on farms.

    Statement 2: Mary lives on a farm.

    Conclusion: Therefore, Mary must be an animal.\,\\\,

  3. Can the statements be valid? Can the conclusions be true?

In the language of logic, there are some things that are always true:

Law of contrapositive

The law of contrapositive states that if p \implies q is true and \sim q is true, then \sim p is true.

For example, consider the statements:

p:Two angles are vertical.
\sim p:Two angles are not vertical.
q:They are congruent.
\sim q:They are not congruent.

Then, in words, by the law of contrapositive:

If two angles are vertical, then they are congruent. \angle A \neq \angle B, therefore \angle A and \angle B are not vertical. We can use the symbol \therefore instead of the word therefore. So, the conclusion of the law of contrapositive could be written:

\angle A \neq \angle B

\therefore \angle A \text{ and } \angle B \text{ are not vertical.}

Statements can also be proven false. A counterexample is used to show an statement is false. A counterexample of a statement confirms the hypothesis but negates the conclusion.

Consider the statement:

All triangles are equilateral.
Triangle with angle measurements of 90, 55, and 35 in degrees.

We could interpret this statement as "If a shape is a triangle, then it is equilateral."

This shape confirms the hypothesis since it is a triangle, but it negates the conclusion since it is not equilateral.

Therefore, this is a counterexample which proves our original statement to be false.

Examples

Example 2

Consider the true conditional statement:

If a shape is a square, then it is a rectangle.
a

Write the converse and determine whether it is true. Provide a counterexample if it is false.

Worked Solution
Create a strategy

The conditional is in the form "if p, then q". Write the converse in the form "if q, then p".

Apply the idea
If a shape is a rectangle, then it is a square.

This statement is false. Consider this rectangle. It is a rectangle, but it is not a square.

A rectangle with length 10 meters and width 5 meters.
b

Write the inverse and determine whether it is true. Provide a counterexample if it is false.

Worked Solution
Create a strategy

The conditional is in the form "if p, then q". Write the inverse in the form "if \sim p, then \sim q".

Apply the idea
If a shape is not a square, then it is not a rectangle.

This statement is false. Consider this rectangle. It is not a square, but it is a rectangle.

A rectangle with length 10 meters and width 5 meters.
c

Write the contrapositive and determine whether it is true. Provide a counterexample if it is false.

Worked Solution
Create a strategy

The conditional is in the form "if p, then q". Write the contrapositive in the form "if \sim q, then \sim p".

Apply the idea
If a shape is not a rectangle, then it is not a square.

This statement is true. Any shape that is not a rectangle cannot be a square since all squares are rectangles.

Reflect and check

We could confirm this by the law of contrapositive. Since the conditional is true and \sim q is true, the contrapositive is also true.

Example 3

Consider the following statements p and q.Consider the conjecture:

Every even number is divisible by 2.

a

Write the contrapositive.

Worked Solution
Create a strategy

First, rewrite the statement in the traditional "if, then" format. Then, write the contrapositive.

Apply the idea

We can rewrite the conjecture as:

"If a number is even, then it is divisible by 2."

So the contrapositive would be:

"If a number is not divisible by 2, then it is not an even number."

b

Use the law of contrapositive to determine if 11 is an even number.

Worked Solution
Create a strategy

The law of contrapositive states that if p \implies q is true and \sim q is true, then \sim p is true.

Apply the idea

Since it is true that if a number is even then it is divisble by 2, p \implies q is true.

Since 11 is not divisible by 2, we know \sim q is true. By the law of contrapositive, \sim p is also true.

Therefore, 11 is not even.

Reflect and check

We may see the word "therefore" replaced by the symbol \therefore

Using this symbol, the last line could have been written:

\therefore \text{ } 11 is not even.

Idea summary

The law of contrapositive states that if p \implies q is true and \sim q is true, then \sim p is true.

A single counterexample can prove that a statement is false.

Outcomes

G.RLT.1

The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.

G.RLT.1a

Translate propositional statements and compound statements into symbolic form, including negations (~p, read “not p”), conjunctions (p ∧ q, read “p and q”), disjunctions (p ∨ q, read “p or q”), conditionals (p → q, read “if p then q”), and biconditionals (p ↔ q, read “p if and only if q”), including statements representing geometric relationships.

G.RLT.1b

Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships.

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