Implication
'If P then Q'. It is false only when P is true and Q is false. In every other case it counts as true.
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Analogy
A promise: 'if it rains, I will bring an umbrella'. You only break it if it rains and you come without one.
Example
x = 4: 16 > 9, so it holds.x = 2: P is false, so no test.A counterexample needs P trueand Q false at once.
In practice
Most theorems have the form 'if assumptions then conclusion', so you must read implications exactly.
Practice
Flashcards
- When is 'if P then Q' false?Only when P is true and Q is false.
- Is the converse of a true implication always true?No. 'If 4 divides n then n is even' is true, but its converse is false for n = 2.
- What does one counterexample show?It disproves a 'for all' claim completely.
Questions people ask
Why is 'if P then Q' true when P is false?
Because the only way to break the promise is P true with Q false. If P never happens, nothing was broken. This is called vacuous truth. For example, 'if 2 is odd then 5 = 7' is a true implication, since the condition fails from the start.
How do you disprove an implication?
Find one counterexample: a case where P is true and Q is false. To disprove 'if n is prime then n is odd', take n = 2. It is prime but even. One counterexample is enough, while examples that fit never prove a general claim.