STEP-BY-STEP LESSON · §2.2
Conditional Statements
Read implication, contraposition, and iff.
Before you begin
Logical Form and Logical Equivalence
p and q stand for statements, each true or false. Compound formulas combine them with logical operations.
p∧q
p∧q is true only when both parts are true. p∨q is true when at least one part is true. ¬p reverses the value.
¬(p∧q)
Symbols
- →
- if… then
- ↔
- if and only if
Definitions and notation for this topic
Implication / if…then
→
False exactly when the hypothesis is true and the conclusion is false.
An implication need not describe a causal connection.
It is false exactly when p is true and q is false. The arrow in f:A→B specifies a function’s source and target, not this truth condition.
Separate glossary example
p → q is false when p=T and q=F.
Biconditional / if and only if
⇔ ↔ iff
True exactly when the two statements have the same truth value.
If and only if requires both directions.
A biconditional is a formula evaluated under a valuation. P≡Q is the claim that formulas P and Q agree under every valuation. The word iff abbreviates if and only if.
Separate glossary example
n is even ⇔ ∃k∈ℤ: n=2k
Contrapositive
(p→q) ≡ (¬q→¬p)
The statement ¬q→¬p is equivalent to p→q and can be proved instead.
For the integer example, prove that an odd n has an odd square. The converse q→p and inverse ¬p→¬q are different statements and are not generally equivalent to p→q.
Separate glossary example
For integer n: if n² is even, then n is even.
If and only if: both directions
A connection requiring both “if P, then Q” and “if Q, then P.”
If gives one direction. Iff includes both, so proving equivalence requires both directions.
Separate glossary example
P iff Q: (P→Q) and (Q→P).
Step by step
Step 1 / 6
Condition and consequence
In p→q, p is the hypothesis and q the conclusion. The implication fails only when p is true and q false.
p→q≡¬p∨q
A false hypothesis does not provide a counterexample to the implication.
Worked example
Compare the implication and its converse: if integer n is divisible by 4, then n is even.
- For the original implication, write n=4k with integer k. Then n=2(2k), which is even because 2k is integer.
- The converse says: if n is even, then n is divisible by 4. Choose n=2: the hypothesis is true but the conclusion is false.
- The contrapositive of the original is: if n is not even, then n is not divisible by 4. It is equivalent to the original, unlike the converse.
Optional self-check
When is p↔q false?
Show answer
When one statement is true and the other false.