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.

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

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.

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

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

b

"If c then d."

Where c and d are both conjectures.

State the converse of the statement.

Worked Solution

c

"If b then d."

b and d are both conjectures.

State the inverse of the statement.

Worked Solution

Idea summary

The

**converse**of "if p then q" is "if q then p" and is logically independent of the originalThe

**inverse**of "if p then q" is "if \sim p then \sim q" and is logically independent of the originalThe

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

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.

*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.\,\\\,*Statement 1:*Only animals live on farms.*Statement 2:*Mary lives on a farm.*Conclusion:*Therefore, Mary must be an animal.\,\\\,Can the statements be valid? Can the conclusions be true?

In the language of logic, there are some things that are always 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. |

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

b

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

Worked Solution

c

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

Worked Solution

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

Every even number is divisible by 2.

a

Write the contrapositive.

Worked Solution

b

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

Worked Solution

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.