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JEE Mains Maths · Mathematical Reasoning

Implications, Converse and Statements in Words

Rewrite an implication in an equivalent form, form its converse and contrapositive, and turn English statements, including 'only if' and 'for all', into symbols.

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Why this matters

Fifteen PYQs, all multiple choice. Eight ask which form is equivalent to a given implication; seven form a converse or contrapositive, or turn an English statement into symbols and negate it. Two ideas cover the page.

Concept 1 of 2: Equivalent forms of an implication

Every implication is ∼A∨B\sim A\vee B, so two implications are equivalent when their or-forms match. From this come the forms JEE uses: the contrapositive, moving a condition from the consequent into the antecedent, and joining two implications that share a consequent or an antecedent.

Definition

  • A→B≡∼A∨B≡∼B→∼AA\rightarrow B\equiv\sim A\vee B\equiv\sim B\rightarrow\sim A (the contrapositive).
  • A→(B→C)≡(A∧B)→CA\rightarrow(B\rightarrow C)\equiv(A\wedge B)\rightarrow C.
  • (A→C)∧(B→C)≡(A∨B)→C(A\rightarrow C)\wedge(B\rightarrow C)\equiv(A\vee B)\rightarrow C.
  • (A→B)∧(A→C)≡A→(B∧C)(A\rightarrow B)\wedge(A\rightarrow C)\equiv A\rightarrow(B\wedge C).

Or-form of an implication

A→B≡∼A∨B≡∼B→∼AA\rightarrow B\equiv\sim A\vee B\equiv\sim B\rightarrow\sim A

Worked example

Which is equivalent to p→(q→r)p\rightarrow(q\rightarrow r): (p∧q)→r(p\wedge q)\rightarrow r or (p→q)→r(p\rightarrow q)\rightarrow r?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 6 April 2023 · Q73Moderate

Example 1 · Mathematical Reasoning · Implications, Converse and Statements in Words

Statement (P⇒Q)∧(R⇒Q)(P \Rightarrow Q) \land (R \Rightarrow Q) is logically equivalent to

An 'or' of implications gives an 'and' on the left

(p→r)∨(q→r)(p\rightarrow r)\vee(q\rightarrow r) is (p∧q)→r(p\wedge q)\rightarrow r, not (p∨q)→r(p\vee q)\rightarrow r. It is ∼p∨∼q∨r\sim p\vee\sim q\vee r, and ∼p∨∼q≡∼(p∧q)\sim p\vee\sim q\equiv\sim(p\wedge q).
Drill 7 more on equivalent forms of an implication

Concept 2 of 2: Converse, contrapositive and statements in words

For p→qp\rightarrow q, the converse swaps the parts, q→pq\rightarrow p. The contrapositive swaps and negates, ∼q→∼p\sim q\rightarrow\sim p, and it is the only one of these equivalent to the original. In words, 'if p then q', 'p only if q' and 'q if p' all mean p→qp\rightarrow q. To negate 'for all', say 'there exists' and negate the condition.

Definition

  • Converse: q→pq\rightarrow p. Inverse: ∼p→∼q\sim p\rightarrow\sim q. Contrapositive: ∼q→∼p\sim q\rightarrow\sim p.
  • 'p only if q' is p→qp\rightarrow q; 'p if and only if q' is p↔qp\leftrightarrow q.
  • The negation of 'for all x, P(x)' is 'there exists x with ∼P(x)\sim P(x)', and the other way round.

The related statements

p→q≡∼q→∼p,q→p≡∼p→∼qp\rightarrow q\equiv\sim q\rightarrow\sim p,\qquad q\rightarrow p\equiv\sim p\rightarrow\sim q

Worked example

Write the converse and the contrapositive of 'If it rains, the match is cancelled.'
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 11 April 2023 · Q161Moderate

Example 2 · Mathematical Reasoning · Implications, Converse and Statements in Words

The converse of the statement ((∼p)∧q)⇒r(( \sim p) \land q) \Rightarrow r is

'Only if' points forward

'p only if q' is p→qp\rightarrow q, not q→pq\rightarrow p. It says p cannot happen without q; it does not say that q forces p.
Drill 6 more on converse, contrapositive and statements in words

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 (2)

  • Equivalent forms of an implication

    Or-form of an implication

    A→B≡∼A∨B≡∼B→∼AA\rightarrow B\equiv\sim A\vee B\equiv\sim B\rightarrow\sim A
  • Converse, contrapositive and statements in words

    The related statements

    p→q≡∼q→∼p,q→p≡∼p→∼qp\rightarrow q\equiv\sim q\rightarrow\sim p,\qquad q\rightarrow p\equiv\sim p\rightarrow\sim q

Watch out for (2)

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