Geometry

Conditional Statements

Big Picture

“If-then” relationships have an important role in geometry. Many geometric statements are actually if-then statements, also called conditional statements.

Key Terms

Conditional Statement: A statement with a hypothesis followed by a conclusion. Can be written in “if-then” form.

Hypothesis: The first, or “if,” part of a conditional statement. An educated guess.

Conclusion: The second, or “then,” part of a conditional statement. The conclusion of a hypothesis.

Converse: A statement where the hypothesis and conclusion of a conditional statement are switched.

Negation (of a statement): The opposite of the original statement.

Inverse: A statement where the hypothesis and conclusion of a conditional statement are negated.

Contrapositive: A statement where the hypothesis and conclusion of a conditional statement are exchanged and negated.

Biconditional Statements: A statement where the original and the converse are both true.

Compound Statement: Combination of two or more statements.

Conjunction:  A compound statement using the word “and.”

Disjunction: A compound statement using the word “or.”

Truth Value: The truth value of a statement is either true or false.

Conditional Statements

Geometry uses conditional statements that can be symbolically written as \(p \rightarrow q\) (read as “if , then”). “If” is the hypothesis, and “then” is the conclusion.

  • The conclusion is sometimes written before the hypothesis.
  • Does not always have to include the words “if” and “then.”

Examples:

  • IF the weather is nice, THEN I’ll wash the car.
  • IF the weather is nice, I’ll wash the car. (“Then” is implied.)
  • I’ll wash the car IF the weather is nice. (The hypothesis comes after the conclusion.)
  • All equiangular triangles are equilateral. (“If” and “then” are implied.)
  • Can be written as: IF the triangle is equiangular, THEN it is equilateral.

The conditional statement is false when the hypothesis is true and the conclusion is false.

  • Only one counter example is needed to prove the conditional statement false.

If the hypothesis is false, the conditional statement is true regardless of whether the conclusion is true or not.

Related Conditionals

The converse of \(p \rightarrow q\) is \(q \rightarrow p\).

The negation of p is “not p,” written as \(\sim p\).

The inverse of \(p \rightarrow q\) is \(\sim p \rightarrow \sim q\).

The contrapositive of \(p \rightarrow q\) is \(\sim{q} \rightarrow \sim{p}\).

Law of Contrapositive

Law of Contrapositive: If p \(p \rightarrow q\) q is true and \(\sim q\) is given, then \(\sim p\) is true.

If the conditional statement is true, the converse and inverse may or may not be true. However, the contrapositive of a true statement is always true. The contrapositive is logically equivalent to the original conditional statement.

  • The converse and inverse are also logically equivalent.

Geometry

Conditional Statements cont.

Biconditional Statements

Biconditional statement: If \(p \rightarrow q\) is true and \(q \rightarrow p\) is true, it can be written as \(p \leftrightarrow q\).

  • The \(\leftrightarrow\) is read as “if and only if” (abbreviated as “iff”).
  • The “if and only if” portion of biconditional statement can be implied.

Compound Statement

A compound statement is a combination of two or more statements. Let p and q each represent a statement.

  • Conjunction: A compound statement that uses the word “and.” \(p \land q\)
  • Disjunction: A compound statement that uses the word “or.” \(p \lor q\)

Truth Table

A truth table can be used to analyze logic problems. It displays the truth values, either true (T) or false (F), for a conditional statement or a compound statement depending on the truth values for the hypothesis and conclusion.

  • Start truth tables with all possible combinations of truth values.
         a. There are four possible combinations for two variables and eight possible combinations for three variables.
  • Do any negations.
  • Do any combinations in parenthesis.
  • Complete the problem.

Conditional

p

q

p \(\rightarrow \) q

T

T

T

T

F

F

F

T

T

F

F

T

Converse

\(p\)

\(q\)

\(q \rightarrow p\)

T

T

T

T

F

T

F

T

F

F

F

T

Conjunction

\(p\)

\(q\)

\(p \wedge q\)

T

T

T

T

F

F

F

T

F

F

F

F

Inverse

\(p\)

\(q\)

\(\sim p\)

\(\sim q\)

\(\sim p \rightarrow \sim q\)

T

T

F

F

T

T

F

F

T

T

F

T

T

F

F

F

F

T

T

T

Contrapositive

\(p\)

\(q\)

\(\sim q\)

\(\sim p\)

\(\sim q \rightarrow \sim p\)

T

T

F

F

T

T

F

T

F

F

F

T

F

T

T

F

F

T

T

T

Disjunction

\(p\)

\(q\)

\(p \vee q\)

T

T

T

T

F

T

F

T

T

F

F

F

Notes