“If-then” relationships have an important role in geometry. Many geometric statements are actually if-then statements, also called conditional statements.
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.
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.
Examples:
The conditional statement is false when the hypothesis is true and the conclusion is false.
If the hypothesis is false, the conditional statement is true regardless of whether the conclusion is true or not.
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: 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.
Biconditional statement: If \(p \rightarrow q\) is true and \(q \rightarrow p\) is true, it can be written as \(p \leftrightarrow q\).
A compound statement is a combination of two or more statements. Let p and q each represent a statement.
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.
p
q
p \(\rightarrow \) q
T
T
T
T
F
F
F
T
T
F
F
T
\(p\)
\(q\)
\(q \rightarrow p\)
T
T
T
T
F
T
F
T
F
F
F
T
\(p\)
\(q\)
\(p \wedge q\)
T
T
T
T
F
F
F
T
F
F
F
F
\(p\)
\(q\)
\(\sim p\)
\(\sim q\)
\(\sim p \rightarrow \sim q\)
T
T
F
F
T
T
F
F
T
T
F
T
T
F
F
F
F
T
T
T
\(p\)
\(q\)
\(\sim q\)
\(\sim p\)
\(\sim q \rightarrow \sim p\)
T
T
F
F
T
T
F
T
F
F
F
T
F
T
T
F
F
T
T
T
\(p\)
\(q\)
\(p \vee q\)
T
T
T
T
F
T
F
T
T
F
F
F