From conditional statements to geometry proofs — master the most tested logic concept.
01
Core Concept
What is a Contrapositive?
A conditional statement has the form If P, then Q. Four related statements can be formed:
Original (Conditional): P → Q
─────────────────────────────────────
Converse: Q → P
Inverse: ¬P → ¬Q Contrapositive: ¬Q → ¬P
KEY FACT: The contrapositive is logically equivalent to the original — always the same truth value.
The converse and inverse are equivalent to each other, but NOT to the original.
02
Memorize These
Original
If P, then Q P → Q
✦ Contrapositive (equivalent)
If ¬Q, then ¬P ¬Q → ¬P
Converse (NOT equivalent)
If Q, then P Q → P
Inverse (NOT equivalent)
If ¬P, then ¬Q ¬P → ¬Q
Equivalency Table
Statement
Form
≡ Original?
Original
P → Q
—
Contrapositive
¬Q → ¬P
✓ YES
Converse
Q → P
✗ NO
Inverse
¬P → ¬Q
✗ NO
03
Worked Examples
Example 1 · Basic Contrapositive
Statement: "If it is raining, then the ground is wet." Write the contrapositive.
Original: raining → ground wet Contrapositive (¬Q → ¬P): ground NOT wet → NOT raining Answer: "If the ground is not wet, then it is not raining."
Example 2 · Logical Equivalence
TRUE statement: "If a polygon is a square, then it has four sides." Which must also be true?
Contrapositive: "If it does not have four sides, then it is not a square." ← TRUE Converse: "If it has four sides, then it is a square." ← NOT necessarily true (could be rectangle) Answer: The contrapositive must be true.
Example 3 · Chained Conditionals
Given: P → Q and Q → R. What is the contrapositive of the combined statement?
Chain: P → Q → R, so P → R (hypothetical syllogism) Contrapositive of (P → R): ¬R → ¬P Answer: "If not R, then not P." (¬R → ¬P)
04
Practice Questions
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