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1. Lines and Angles
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Grades 9-12

1.02 Definitions and conditional statements

Lesson
Practice

Introduction

We will get an introduction to conditional statements in this lesson. We will write and analyze the truth of conditional and biconditional statements to help draw logical conclusions. With that information, we will learn about other ways to relate, combine, and change conjectures.

Conditional and biconditional statements

To describe how one event can lead to another, we can use conditional statements to connect a hypothesis to a conclusion.

Conditional statement

A logical statement that joins a hypothesis to a conclusion. Conditional statements can be written in the form "if (hypothesis), then (conclusion)".

Example:

If today is Monday, then tomorrow is Tuesday.

Hypothesis

The condition within a conditional statement, usually represented as the "if" part.

Example:

today is Monday...

Conclusion

The consequence of a conditional statement, usually represented as the "then" part.

Example:

...tomorrow is Tuesday

A conditional statement itself has a truth value:

p \text{ (hypothesis)}q \text{ (conclusion)}p \to q
TrueTrueTrue
TrueFalseFalse
FalseTrueTrue
FalseFalseTrue

If p being true leads to q being true, then the conditional statement is true.

If p being true can lead to q being false, however, then the conditional statement is false.

Note that a conditional statement claims nothing if the hypothesis is false. In such a case the conditional statement is considered to be true.

In a situation where we have the hypothesis leading to a conclusion and the conclusion leading back to the hypothesis, we can form a biconditional statement.

Biconditional statement

A logical statement that joins a hypothesis and a conclusion in both directions. Biconditional statements can be written in the form "(hypothesis) if and only if (conclusion)."

Example:

Two adjacent angles are a linear pair if and only if they form a straight line.

A biconditional statement is equivalent to a conditional statement p \to q and its converse q \to p simultaneously. Symbolically, we represent this biconditional statement asp \longleftrightarrow q

If a biconditional statement is true, then we can write it as a definition.

Examples

Example 1

Consider the conditional statement "If a person is French, then they are European."

a

State the hypothesis and conclusion of the statement.

Worked Solution
Create a strategy

The hypothesis is the condition represented as the "if" part, while the conclusion is the consequence represented by the "then" part.

Apply the idea

The hypothesis of the statement is "a person is French."

The conclusion of the statement is "they are European."

b

Determine the truth value of the conditional statement.

Worked Solution
Create a strategy

If p being true leads to q being true, then the conditional statement is true.

If p being true can lead to q being false, however, then the conditional statement is false.

Note that a conditional statement claims nothing if the hypothesis is false. In such a case the conditional statement is considered to be true.

Apply the idea

Since France is a country in Europe, it is a true statement that if a person is French, then they are European. The conditional statement is true.

Reflect and check

A conditional statement that would be false is "If a person is Argentinian, then they are European" because although a person may be Argentinian, the country of Argentina is not located in Europe.

Example 2

Consider the biconditional statement "An animal is a bird if and only if it can fly."

a

State the two conditional statements that this biconditional statement represents.

Worked Solution
Create a strategy

The two parts of the biconditional statement are "is a bird" and "it can fly". These two parts will be the hypotheses and conclusions of our two statements.

Apply the idea

The two conditional statements represented by the biconditional statement are:

  • If an animal is a bird, then it can fly.

  • If an animal can fly, then it is a bird.

Reflect and check

Notice that these two conditional statements are converses of one another.

b

Determine whether the biconditional statement is true or false. If it is false, give a counterexample.

Worked Solution
Create a strategy

To check if the biconditional statement is true or false, we can check whether the two statements it represents are true or false.

Apply the idea

The statement "If an animal is a bird, then it can fly" is false. We can give the counterexample of a penguin, which is a bird that cannot fly.

The statement "If an animal can fly, then it is a bird" is also false. We can give the example of a bat, which can fly but is not a bird.

Since we do not have both statements being true, the biconditional statement is false.

Reflect and check

Only one of the conditional statements needed to be false for the biconditional statement to be false.

Example 3

Determine whether each of the following is a valid definition or not:

a

"A midpoint is a point that cuts a line segment into two congruent segments."

Worked Solution
Create a strategy

This statement is a valid definition if it can be written as a true biconditional statement.

Apply the idea

We can write this statement as the biconditional statement "A point is a midpoint if and only if it cuts a line segment into two congruent segments."

This biconditional statement represents the two statements:

  • If a point is a midpoint, then it cuts a line segment into two congruent segments.
  • If a point cuts a line segment into two congruent segments, then it is the midpoint.

Both of these statements are true, so the biconditional statement is true.

Since we can write the given statement as a true biconditional statement, it is a valid definition.

b

"If an angle is acute, then its measure is less than 90\degree."

Worked Solution
Create a strategy

We can first write the statement as a biconditional statement. Then we can try to find a counterexample to disprove the statement.

Apply the idea

The biconditional statement would be "An angle is acute if and only if its measure is less than 90\degree."

This represents two statements:

  • "If an angle is acute, then its measure is less than 90\degree.
  • If an angle's measure is less than 90\degree then it is acute.

We can disprove the second statement with an angle measuring 0 \degree because it is less than 90 \degree but it is not acute.

c

"A triangle is a polygon with three sides."

Worked Solution
Apply the idea

We can write this statement as the biconditional statement "A polygon is a triangle if and only if it has three sides."

This biconditional statement represents the two statements:

  • If a polygon is a triangle, then it has three sides.
  • If a polygon has three sides, then it is a triangle.

Both of these statements are true, so the biconditional statement is true.

Since we can write the given statement as a true biconditional statement, it is a valid definition.

Reflect and check

If either one of the conditional statements was false, then it would not be a valid definition.

Idea summary

Recall the forms of conditional and biconditional statements:

  • Conditional statement: "If (hypothesis), then (conclusion)"
  • Biconditional statement: A situation where the hypothesis leads to a conclusion and the conclusion leads back to the hypothesis
  • Definition: A biconditional statement that is true

Inverse, converse, and contrapositive of a conditional statement

Consider the following ways to relate, combine, and change conjectures that may change the truth value of statements:

Negation

The negative of a statement, represented by the symbol \sim

Consider the following statement and its negation:

Statement \left(P\right): Birds eat elephants.

Negation (Not P): Birds do not eat elephants.

Symbolic: \sim P

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

Contrapositive

A statement that negates both parts of a conditional statement (like for the inverse), and also switches the order (like for the converse)

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

Examples

Example 4

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

Derived logical statements:

  • The negation of P is \sim P, and has opposite truth value to P
  • 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

Outcomes

G.CO.A.1

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

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