What it is
An implication A ⇒ B reads “if A then B”. It claims that every time A holds, B holds too. It says nothing about what happens when A does not hold.
- x = 3 ⇒ x² = 9. True: 3² = 9.
- n is divisible by 6 ⇒ n is divisible by 3. True: if n = 6k, then n = 3 · 2k.
The reverse, B ⇒ A, is called the converse, and it need not hold.
- x² = 9 ⇒ x = 3 is false. x = −3 makes the left side true and the right side false. Such a case is a counterexample, and a single one is enough to show an implication false.
- n is divisible by 3 ⇒ n is divisible by 6 is false. n = 9 is the counterexample.
An equivalence A ⇔ B holds when the implication goes both ways. It reads “A exactly when B” or “A if and only if B”.
- x + 2 = 5 ⇔ x = 3. To the right: subtract 2. To the left: add 2.
- A triangle with sides a, b, c where c is longest is right-angled ⇔ a² + b² = c². One direction is the Pythagorean theorem, the other is its converse.
- n is even ⇔ n is divisible by 2.
Not an equivalence: x² = 9 ⇔ x = 3, because it only goes one way.
Why it matters
When you solve an equation you write line after line. Each step is either an equivalence or only an implication, and that decides whether the answer must be checked.
- In a linear equation every step is an equivalence: 3x + 5 = 17 ⇔ 3x = 12 ⇔ x = 4. The solutions stay the same throughout.
- Squaring both sides is only an implication. Take √(x + 2) = x. Squaring gives x + 2 = x², so (x − 2)(x + 1) = 0 and x = 2 or x = −1. Check: √4 = 2 holds, but √1 = 1 is not −1. That is why the roots must be checked in radical equations.
- That the Pythagorean theorem and its converse both hold is what lets you use it to test for a right angle, not just to find a side.
The same thinking applies outside mathematics. “All dogs are animals” does not hold the other way round.
Common misconceptions
- “A ⇒ B and B ⇒ A are the same.” No. x = 3 ⇒ x² = 9 holds; x² = 9 ⇒ x = 3 does not.
- “The arrow ⇒ is an equals sign.” An equals sign goes between numbers or expressions. The arrow goes between statements.
- “Many examples show that an implication holds.” They do not. The expression n² + n + 41 gives a prime for n = 0, 1, …, 39, but for n = 40 it is 1681 = 41², which is not prime.
- “Squaring both sides is an equivalent step.” It is only an implication, so false roots may have crept in. That is why the answers are checked.
Common questions
What is the difference between ⇒ and ⇔?
⇒ goes one way: from what holds to what follows. ⇔ goes both. A ⇔ B means the same as both A ⇒ B and B ⇒ A
holding.
What does “if and only if” mean?
The same as ⇔. “A number is even if and only if it is divisible by 2” means it holds both ways: even numbers
are divisible by 2, and numbers divisible by 2 are even.
How do you show that an implication is false?
With a counterexample: a case where the first statement holds and the second does not. For “x² = 9 ⇒ x = 3”,
x = −3 is enough.