Logical Reasoning
How to Write the converse of an implication
Use this to form the converse of an “if … then …” statement and judge whether it is true.
Before you start
- Know what an implication (an 'if … then …' statement) is.
- Be able to identify the antecedent (the 'if' part) and the consequent (the 'then' part).
- Understand that one counter-example is enough to show a statement is false.
When to use it
Use this to form the converse of an “if … then …” statement and judge whether it is true.
The steps
- Identify the antecedent (the “if” part) and the consequent (the “then” part).
- Swap them: the consequent becomes the new “if”, the antecedent the new “then”.
- Write the converse as a full statement.
- Test whether the converse is true, it may be true or false independently of the original.
- Give a counter-example if it is false.
Worked example
Write the converse of the statement 'If a number is a multiple of 4, then it is a multiple of 2', and decide whether the converse is true.
- Identify the parts: the antecedent (if) is 'a number is a multiple of 4'; the consequent (then) is 'it is a multiple of 2'.
- Swap them: the consequent becomes the new 'if', and the antecedent becomes the new 'then'.
- Write the converse as a full statement: 'If a number is a multiple of 2, then it is a multiple of 4'.
- Test whether the converse is true: a multiple of 2 need not be a multiple of 4, so it is not always true.
- Give a counter-example: 6 is a multiple of 2 but not a multiple of 4, so the converse is false.
A second example, with a twist
Here the converse turns out to be true, so step 5 gives no counter-example, unlike the first example. Write the converse of 'If x + 3 = 7, then x = 4', and decide whether the converse is true.
- Identify the parts: the antecedent (if) is 'x + 3 = 7'; the consequent (then) is 'x = 4'.
- Swap them: 'x = 4' becomes the new 'if' and 'x + 3 = 7' becomes the new 'then'.
- Write the converse as a full statement: 'If x = 4, then x + 3 = 7'.
- Test whether the converse is true: substitute x = 4 into x + 3, giving 4 + 3 = 7, which is correct, so the converse is true.
- Because the converse is true, no counter-example exists, you cannot find a value of x that makes the 'if' true but the 'then' false.
Practise this in a KBAT problem
Frequently asked questions
If the original statement is true, is the converse also true?
Not necessarily. The converse is a completely separate statement, so it can be true or false no matter what the original is.
'If x is a multiple of 4 then x is even' is true, but its converse is false. Always test the converse on its own before deciding.
What exactly is a counter-example?
A counter-example is a single case where the 'if' part is true but the 'then' part is false. Finding just one is enough to prove a statement false.
For 'if x is even then x is a multiple of 4', the number 6 is even but not a multiple of 4, so it is a counter-example.
How is the converse different from the negation?
They are different operations. The converse swaps the antecedent and consequent, keeping both positive: 'if p then q' becomes 'if q then p'.
The negation denies the statement instead, saying it is not true. In SPM you must know which one the question asks for, because the answers are not the same.