The learning goal
For p→q, the converse is q→p, the inverse is not-p→not-q and the contrapositive is not-q→not-p. The original and its contrapositive are logically equivalent. A true original does not automatically establish its converse; logical equivalence compares all truth cases.
Keep p and q fixed
Our domain is the positive integers. Let p mean n is a multiple of 4 and q mean n is even. The OpenStax equivalent-statements chapter gives the logical forms and equivalence relationships; the integer exercise below is independently constructed for this article.
The original has a general proof: if n=4k, then n=2(2k), so n is even. This explains every qualifying integer rather than declaring a universal result after listing 4,8 and 12.
| Name | Form | Statement in this exercise |
|---|---|---|
| Original | p→q | Multiple of 4 implies even |
| Converse | q→p | Even implies multiple of 4 |
| Inverse | not-p→not-q | Not a multiple of 4 implies not even |
| Contrapositive | not-q→not-p | Not even implies not a multiple of 4 |
Why 6 exposes two mistakes
For n=6, q is true and p is false. The converse has a true premise and a false conclusion, so it fails. For the inverse, not a multiple of 4 is true while not even is false. The same 6 therefore rejects the inverse as well.
It does not reject the original because 6 does not satisfy the original’s premise. To refute every p is q, find p true and q false rather than supplying an object outside p. See what one counterexample can disprove for the role of quantifiers.
A counterexample must fit the target premise
6 is not a multiple of 4, so it disproves that all multiples of 4 are even.
6 is even but not a multiple of 4, so it disproves the converse.
Compare complete columns, not one matching case
Material implication p→q is false only when p is true and q false. The contrapositive not-q→not-p is false on exactly that same row, so its entire column matches. The converse is false when p is false and q true, producing a different column.
The table lists abstract logical possibilities. It does not claim each row can occur among our integers. In fact, the proof rules out a multiple of 4 that is not even. A truth table is not a list of observed numerical records.
| p,q | p→q | not-q→not-p | q→p |
|---|---|---|---|
| T,T | T | T | T |
| T,F | F | F | T |
| F,T | T | T | F |
| F,F | T | T | T |
A contrapositive proof retains the direction
Suppose n is not even, but assume it is a multiple of 4. Then n=4k=2(2k), making it even, a contradiction. Thus it cannot be a multiple of 4. The contrapositive retains the original constraint rather than converting a sufficient condition into a necessary one.
Reading the original as being a multiple of 4 is sufficient for being even can help. The terminology is not a replacement for reasoning. Remembering only add a negation can produce not-p→not-q and omit the essential swap.
Transfer the structure from 4 and 2 to 6 and 3
Stay within positive integers and use the original if n is a multiple of 6, then n is a multiple of 3. Hide the answer, write its three variants, identify equivalents and find an object rejecting each false variant.
Reference answer: the converse is multiple of 3 implies multiple of 6, rejected by 9. The inverse is not a multiple of 6 implies not a multiple of 3, also rejected by 9. The contrapositive is not a multiple of 3 implies not a multiple of 6, equivalent to the original. The original follows from 6k=3(2k). Changing the values while preserving the structure tests understanding beyond memorizing one example.
Classify the first error
Direction
The converse was substituted for the original: revisit which part is p and which is q.
Negation
Negations were added without swapping: inspect not-q→not-p.
Counterexample
The object does not satisfy the target premise: mark its truth values again.
Record the first incorrect transformation
Do not copy only the slogan that a contrapositive is equivalent. Save the original, the meanings of p and q, your attempted form, the first swap or negation error and an object that checks it. Use examples, non-examples and boundaries to progress beyond divisibility exercises.
This article concerns conditionals in propositional logic. Everyday causal, promised or statistical statements do not automatically behave as material implication. If a sentence says usually, possibly or only some, clarify its quantifiers and context before applying the same transformation.
Explain these points independently
- The domain and meanings of p and q remain fixed.
- Converse, inverse and contrapositive are written correctly.
- A counterexample makes the target premise true and conclusion false.
- Equivalence is explained by a complete matching truth column, not a single example.
Common questions
If a converse happens to be true, is it logically equivalent?
Both being true in a particular relationship does not establish equivalence of the general logical forms. A genuine two-way relationship needs its own proof; form equivalence compares every truth case.
Why is a conditional not false whenever its premise is false?
Material implication is defined to be false only for a true premise and false conclusion. It constrains a forbidden combination rather than measuring how often a premise occurs.
Can several matching integers prove the original?
They can illustrate it but cannot prove a universal statement over positive integers. Here the general argument is n=4k=2(2k).
