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MHT-CET Maths · Mathematical Logic

Converse, Inverse and Contrapositive

Every conditional has three relatives — swap the parts for the converse, negate both for the inverse, do both for the contrapositive — and only the contrapositive is equivalent to the original.

All Mathematical Logic notesMHT-CET Maths strategy

Why this matters

This is one of the chapter's larger subtopics at 28 PYQs and a moderate 21% HARD, and it is almost pure pattern work once the three forms are automatic. The paper tests it in two registers: half the questions hand you a symbolic pattern, half hand you an English sentence about triangles, judges or integers. The HARD ones stack the operations — the negation of the contrapositive, or the contrapositive of the inverse — which is only bookkeeping if you apply one rule at a time.

Concept 1 of 5: The Three Relatives of a Conditional

Start from 'if p then q'. Swapping the two halves gives the converse; negating both halves gives the inverse; doing both gives the contrapositive. Two operations, applied separately or together, generate all three.

Definition

From the original p→qp \to q:

  • Converse q→pq \to p — swap only.
  • Inverse ∼p→∼q\sim p \to \sim q — negate only.
  • Contrapositive ∼q→∼p\sim q \to \sim p — swap and negate.

Note the shape: all three are still conditionals. If your answer has lost its arrow, you have negated rather than transformed.

FormSymbolicBuilt byEquivalent to original?
Originalp→qp \to q—Yes, trivially
Converseq→pq \to pSwap the two partsNo
Inverse∼p→∼q\sim p \to \sim qNegate both parts, keep the orderNo
Contrapositive∼q→∼p\sim q \to \sim pSwap and negate bothYes — always
The only equivalent relative, and the one the paper asks about most. A statement and its contrapositive always share a truth value.
Converse and inverseq→pq \to p and ∼p→∼q\sim p \to \sim qEach is the contrapositive of the otherEquivalent to EACH OTHER, not to the original
Memorise the last column. Most option lists contain all three relatives, so knowing the forms is not enough — you must know which one is being asked for.
Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 11th May Shift 2 · Q105Easy

Example 1 · Mathematical Logic · Converse, Inverse, and Contrapositive

Contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is

Offering the converse where the contrapositive was asked

Both are single-step transformations of the same statement and both appear in the options every time. Read the question word again before you commit: 'converse' swaps, 'contrapositive' swaps AND negates.
Drill 6 more on the three relatives of a conditional

Concept 2 of 5: Only the Contrapositive Shares the Truth Value

'If it is a dog, then it is an animal' is true, and so is 'if it is not an animal, then it is not a dog'. But 'if it is an animal, then it is a dog' is plainly false. That is the whole content of this rule, and it is worth carrying as a concrete example.

Definition

p→q≡  ∼q→∼pp \to q \equiv \;\sim q \to \sim p, and this is the only equivalence among the four forms.

  • The converse and the inverse are equivalent to each other, and to neither the original nor the contrapositive.
  • Practical use: to find the truth value of a conditional, you may compute its contrapositive's truth value instead — they must match.
  • Practical use: a statement can be replaced by its contrapositive in any argument without changing anything.

The equivalence pairs

p→q≡  ∼q→∼pq→p≡  ∼p→∼qp→q≢q→pp \to q \equiv\; \sim q \to \sim p \qquad q \to p \equiv\; \sim p \to \sim q \qquad p \to q \not\equiv q \to p
  • ≡\equivlogically equivalent — identical last column

Worked example

Let pp: 'If 7 is an odd number then 7 is divisible by 2', and let qq: 'If 7 is a prime number then 7 is an odd number'. Find the truth values of the contrapositives of pp and qq.
Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 12th May Shift 2 · Q150Easy

Example 2 · Mathematical Logic · Converse, Inverse, and Contrapositive

Let Statement 1: If a quadrilateral is a square, then all of its sides are equal. Statement 2: All the sides of a quadrilateral are equal, then it is a square.

Assuming the converse follows from the original

'If a number is a multiple of 9 then it is a multiple of 3' is true; its converse is false (take 6). A true conditional says nothing whatever about its converse, and stems are built around exactly this gap.
Drill 2 more on only the contrapositive shares the truth value

Concept 3 of 5: Converting to Conditional Form First

Several stems ask for the contrapositive of something that is not written as a conditional at all — it arrives as an OR. You cannot swap and negate parts that are not there, so rewrite it as an arrow first using the conditional law.

Definition

Use p→q≡  ∼p∨qp \to q \equiv \;\sim p \vee q in reverse to expose a hidden conditional:

  • ∼p∨q\sim p \vee q is already p→qp \to q.
  • p∨qp \vee q is ∼p→q\sim p \to q — negate the part you move to the front.
  • q∨∼pq \vee \sim p is p→qp \to q as well, since a disjunction may be read in either order.

Only once the statement is an arrow can you take its converse, inverse or contrapositive.

Conditional law, used in reverse

∼p∨q≡p→qp∨q≡  ∼p→qp→q≡  ∼q→∼p\sim p \vee q \equiv p \to q \qquad p \vee q \equiv\; \sim p \to q \qquad p \to q \equiv\; \sim q \to \sim p
  • ∨\veethe disjunction hiding a conditional

Worked example

Find the contrapositive of ∼p∨(q∧∼r)\sim p \vee (q \wedge \sim r).
Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 23 April Shift I · Q137Moderate

Example 3 · Mathematical Logic · Converse, Inverse, and Contrapositive

The contrapositive of the statement ∼p∨(q∧∼r)\sim p\vee (q\land \sim r) is

Negating the wrong part when converting an OR

p∨qp \vee q becomes ∼p→q\sim p \to q, not p→qp \to q. The letter that moves into the antecedent position picks up a negation — check it against the law rather than by feel.
Drill 1 more on converting to conditional form first

Concept 4 of 5: Stacked Operations: Negation of a Contrapositive, Contrapositive of an Inverse

The HARD questions here just chain two rules you already have. There is no new idea — only the discipline of doing one operation, writing the result down, and then starting the second from that written result rather than from the original.

Definition

Work strictly left to right through the phrase, innermost operation first:

  • 'the negation of the contrapositive of S' means: form the contrapositive of S, then negate it.
  • 'the contrapositive of the inverse of S' means: form the inverse, then take its contrapositive.

A useful shortcut: the contrapositive of the inverse is the converse, because inverse and converse are already contrapositives of each other. So that particular chain collapses to one step.

The chain that collapses

contrapositive(inverse(p→q))≡q→p∼(contrapositive(p→q))≡  ∼q∧p\text{contrapositive}(\text{inverse}(p \to q)) \equiv q \to p \qquad \sim(\text{contrapositive}(p \to q)) \equiv\; \sim q \wedge p
  • inverse∼p→∼q\sim p \to \sim q
  • contrapositiveswap and negate

Worked example

Find the negation of the contrapositive of p→(∼q∧r)p \to (\sim q \wedge r).
Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 2nd May Shift 2 · Q142Hard

Example 4 · Mathematical Logic · Converse, Inverse, and Contrapositive

The negation of contrapositive of the statement p→(¬q∧r)p\to(\neg q \wedge r) is

Applying the second operation to the original statement

'The negation of the contrapositive' does not mean 'the negation, and also the contrapositive, of the original'. Form the contrapositive, write it down, and treat THAT as the new statement. Skipping the written intermediate is where these go wrong.
Drill 3 more on stacked operations: negation of a contrapositive, contrapositive of an inverse

Concept 5 of 5: Necessary and Sufficient Condition Language

The same conditional can be dressed in several English phrasings, and the paper uses them interchangeably. 'p only if q' sounds as though q comes first, but it is still 'if p then q' — the word 'only' is what reverses the reading.

Definition

All of the following say p→qp \to q:

  • if p, then q
  • p only if q
  • q is necessary for p
  • p is sufficient for q
  • ∼q→∼p\sim q \to \sim p (the contrapositive)

And p↔qp \leftrightarrow q is read 'p if and only if q', equivalently 'p is necessary and sufficient for q'.

Equivalent phrasings of one conditional

p→q  ≡  (p only if q)  ≡  (q necessary for p)  ≡  (p sufficient for q)  ≡  ∼q→∼pp \to q \;\equiv\; (p \text{ only if } q) \;\equiv\; (q \text{ necessary for } p) \;\equiv\; (p \text{ sufficient for } q) \;\equiv\; \sim q \to \sim p
  • necessarythe CONSEQUENT — it must hold for the antecedent to
  • sufficientthe ANTECEDENT — it is enough to guarantee the consequent

Worked example

Let p: a number is a multiple of 9, and q: a number is a multiple of 3. Which of these say the same thing as 'if p then q'? (i) p only if q. (ii) q is necessary for p. (iii) q only if p.
Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2021 · May Shift 1 · Q114Moderate

Example 5 · Mathematical Logic · Converse, Inverse, and Contrapositive

Consider the following statements: r: If a number is a multiple of 9, then it is a multiple of 3. Let pp: A number is a multiple of 9, and qq: A number is a multiple of 3. Then, 'if p then q' is the same as

Reading 'p only if q' as 'if q then p'

The word order tempts you to put q first, but 'p only if q' is p→qp \to q — q is the NECESSARY condition, so it sits as the consequent. 'If q then p' would be the converse, and it is the distractor supplied.

Swapping necessary and sufficient

Necessary is the consequent; sufficient is the antecedent. A quick anchor: being a multiple of 3 is NECESSARY for being a multiple of 9, and being a multiple of 9 is SUFFICIENT for being a multiple of 3.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

  • Only the Contrapositive Shares the Truth Value

    The equivalence pairs

    p→q≡  ∼q→∼pq→p≡  ∼p→∼qp→q≢q→pp \to q \equiv\; \sim q \to \sim p \qquad q \to p \equiv\; \sim p \to \sim q \qquad p \to q \not\equiv q \to p
  • Converting to Conditional Form First

    Conditional law, used in reverse

    ∼p∨q≡p→qp∨q≡  ∼p→qp→q≡  ∼q→∼p\sim p \vee q \equiv p \to q \qquad p \vee q \equiv\; \sim p \to q \qquad p \to q \equiv\; \sim q \to \sim p
  • Stacked Operations: Negation of a Contrapositive, Contrapositive of an Inverse

    The chain that collapses

    contrapositive(inverse(p→q))≡q→p∼(contrapositive(p→q))≡  ∼q∧p\text{contrapositive}(\text{inverse}(p \to q)) \equiv q \to p \qquad \sim(\text{contrapositive}(p \to q)) \equiv\; \sim q \wedge p
  • Necessary and Sufficient Condition Language

    Equivalent phrasings of one conditional

    p→q  ≡  (p only if q)  ≡  (q necessary for p)  ≡  (p sufficient for q)  ≡  ∼q→∼pp \to q \;\equiv\; (p \text{ only if } q) \;\equiv\; (q \text{ necessary for } p) \;\equiv\; (p \text{ sufficient for } q) \;\equiv\; \sim q \to \sim p

Reference tables (1)

The Three Relatives of a Conditional5 rows
FormSymbolicBuilt byEquivalent to original?
Originalp→qp \to q—Yes, trivially
Converseq→pq \to pSwap the two partsNo
Inverse∼p→∼q\sim p \to \sim qNegate both parts, keep the orderNo
Contrapositive∼q→∼p\sim q \to \sim pSwap and negate bothYes — always
The only equivalent relative, and the one the paper asks about most. A statement and its contrapositive always share a truth value.
Converse and inverseq→pq \to p and ∼p→∼q\sim p \to \sim qEach is the contrapositive of the otherEquivalent to EACH OTHER, not to the original
Memorise the last column. Most option lists contain all three relatives, so knowing the forms is not enough — you must know which one is being asked for.

Watch out for (6)

Test yourself on Mathematical Logic

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