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1. Foundations of Geometry
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United States of AmericaFL
Grade 912

1.05 Definitions and conditionals

Lesson
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Lesson

Concept summary

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

We can express conditional statements in a compact, algebraic form using symbols.

\displaystyle p \to q
\bm{p}
is the hypothesis
\bm{q}
is the conclusion
\bm{\to}
is the symbol showing that p leads to q

p \to q can be read as "if p then q", or "p implies q".

Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion of that statement is also true.

In symbols, if p \to q is true and p is true, then q is also true.

Law of Syllogism

If the conclusion of a true conditional statement is the hypothesis of another true conditional statement, then we can join the two statements.

In symbols, if p \to q and q \to r are both true, then p \to r is also true.

A number of related types of statements to a conditional statement can be obtained by switching the hypothesis and conclusion, and by considering the negation of these statements.

Negation

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

Example:

If p is "today is Monday", then the negation \sim p is "today is not Monday".

Converse statement

A statement formed by switching the hypothesis and conclusion of a conditional statement. Symbolically, this is q \to p.

Inverse statement

A statement formed by negating the hypothesis and conclusion of a conditional statement. Symbolically, this is \sim p \to \sim q.

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.

Definition

A statement that describes a specific (mathematical) object and can be written as a true biconditional statement.

Example:

A linear pair is two adjacent angles that form a straight line.

pqp \longleftrightarrow q
TrueTrueTrue
TrueFalseFalse
FalseTrueFalse
FalseFalseTrue

A biconditional statement is only true if both the conditional and converse statements it represents are true. If either is false, then the biconditional statement is also false.

Note, however, that a biconditional statement claims nothing if both the hypothesis and conclusion are false.

Worked examples

Example 1

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.

Approach

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.

Solution

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.

Reflection

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.

Approach

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

Solution

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.

Reflection

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

Example 2

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."

Approach

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

Solution

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."

Solution

Notice that this is a conditional statement, so it cannot be written as an "if and only if" statement.

This means that we cannot write it as biconditional statement and so it cannot be a valid definition.

c

"A triangle is a polygon with three sides."

Solution

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.

Reflection

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

Outcomes

MA.912.LT.4.3

Identify and accurately interpret "if... then", "if and only if", "all" and "not" statements. Find the converse, inverse and contrapositive of a statement.

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