An implication has the form “if p, then q”. Its converse swaps the two parts to “if q, then p”, and a true implication can have a false converse.
This lesson continues negating statements with quantifiers in the logical reasoning chapter. The same truth-value checking habit carries over.
What are the four forms of an implication?
Start from “if p, then q”, then swap or negate the parts. Each form has its own name.
| Form | Pattern |
|---|---|
| Implication | if p, then q |
| Converse | if q, then p |
| Inverse | if not p, then not q |
| Contrapositive | if not q, then not p |
The contrapositive always matches the original in truth value. The converse and inverse match each other, but not necessarily the original.
Worked example: multiples of 9
Let p be “a number is a multiple of 9” and q be “a number is a multiple of 3”.
Implication: If a number is a multiple of 9, then it is a multiple of 3. This is true, because 9k = 3 × 3k.
Converse: If a number is a multiple of 3, then it is a multiple of 9. This is false, since 12 is a multiple of 3 but 12 ÷ 9 is not a whole number.
Inverse: If a number is not a multiple of 9, then it is not a multiple of 3. This is false for the same number, 12.
Contrapositive: If a number is not a multiple of 3, then it is not a multiple of 9. This is true. A number that is not a multiple of 3 cannot be 9k, because 9k is always a multiple of 3.
In this example the implication and its contrapositive are true, while the converse and inverse are false.
The mistake that costs marks
The common slip is to assume the converse is true because the original is. It feels natural, since the two sentences use the same words.
| Step | Wrong | Right |
|---|---|---|
| Original | If x = 4, then x² = 16 | If x = 4, then x² = 16 |
| Converse | If x² = 16, then x = 4 | If x² = 16, then x = 4 |
| Truth value of the converse | true, because the original is | false, since x = −4 also gives 16 |
One counterexample, x = −4, is enough to show the converse is false.
Check yourself
Write the converse of “If a shape is a square, then it has four sides”. Decide whether the original and the converse are true, and give a counterexample if one is needed.
Answer
The original is true, because every square has four sides.
The converse is “If a shape has four sides, then it is a square.” This is false. A rectangle with sides 3 cm and 5 cm has four sides and is not a square, so it is a counterexample.
What to study next
Next, test deductive arguments with counterexamples. If you want to see the forms applied to exam-style wording, use the logical reasoning practice set.
Record each implication you mix up in a mistake log. For a teacher to check your reasoning as you go, see online one-to-one Mathematics tuition.