Logical Reasoning · 3.1.5
Converse, inverse, contrapositive
Given an implication 'if p then q', students form its converse ('if q then p'), inverse ('if not p then not q'), and contrapositive ('if not q then not p'), then compare truth values. A key result is that an implication and its contrapositive always share the same truth value.
The official learning standard (3.1.5)
“Construct and compare the truth value of converse, inverse and contrapositive of an implication. Mathematical statements need to be emphasised.”
What it means
Given an implication 'if p then q', students form its converse ('if q then p'), inverse ('if not p then not q'), and contrapositive ('if not q then not p'), then compare truth values. A key result is that an implication and its contrapositive always share the same truth value.
How it is examined
Paper 1 typically gives an implication and asks students to write its converse, inverse, or contrapositive, or to state the truth value of each. Paper 2 may require comparing truth values across all four statements and explaining the relationship between an implication and its contrapositive.
Worked example
Given the implication 'If x = 3, then x² = 9', write and determine the truth value of its converse, inverse, and contrapositive.
- Original implication: 'If x = 3, then x² = 9' TRUE
- Converse (swap p and q): 'If x² = 9, then x = 3' FALSE, since x could be -3
- Inverse (negate both, keep order): 'If x ≠ 3, then x² ≠ 9' FALSE, since x = -3 gives x² = 9
- Contrapositive (negate and swap): 'If x² ≠ 9, then x ≠ 3' TRUE
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I form the converse, inverse, and contrapositive from 'if p then q'?
Converse swaps the two parts: 'if q then p'. Inverse negates both parts but keeps the order: 'if not p then not q'.
Contrapositive negates both parts and swaps them: 'if not q then not p'.
Why do an implication and its contrapositive always match in truth value?
They express the same logical relationship from opposite directions, saying 'p forces q' is logically identical to saying 'not q forces not p'. This equivalence is a standard result that always holds, regardless of what p and q represent.
If the original implication is true, must the converse also be true?
Not necessarily. The converse is a different statement and can be true or false independently, for example 'if x = 3 then x² = 9' is true, but its converse 'if x² = 9 then x = 3' is false since x could be -3.