NCERT Solutions & Concepts for Class 11 Mathematical Reasoning Exercise 14.4

Understanding Implications: 'If-Then' Statements

An implication is a compound statement formed by connecting two simpler statements $p$ and $q$ using the connective 'if... then...'. We write this logically as $p \implies q$, which reads as 'p implies q'. Here, $p$ is called the hypothesis (or antecedent/sufficient condition), and $q$ is called the conclusion (or consequent/necessary condition). In mathematical reasoning, expressing this implication correctly in its different equivalent forms is vital. Specifically, the statement $p \implies q$ can be written in five equivalent ways: (1) If $p$ then $q$, (2) $p$ implies $q$, (3) $p$ is sufficient for $q$, (4) $q$ is necessary for $p$, and (5) $p$ only if $q$. Understanding that 'p only if q' means 'if p, then q' is one of the most common hurdles for students, as it sounds counter-intuitive at first. Let us demystify these forms so you can write them accurately in your exams.

Key Logical Terms Explained

Implication (Conditional Statement)

A compound statement of the form 'If p, then q', denoted by $p \implies q$.

Contrapositive

The logical statement formed by negating and reversing the terms of an implication. The contrapositive of 'If p, then q' is 'If not q, then not p' ($\sim q \implies \sim p$).

Converse

The logical statement formed by swapping the order of the hypothesis and conclusion. The converse of 'If p, then q' is 'If q, then p' ($q \implies p$).

Biconditional (If and Only If)

A compound statement that is true when both the implication ($p \implies q$) and its converse ($q \implies p$) are simultaneously true. It is written as $p \iff q$.

How to Construct the Converse and Contrapositive

1. Identify Component Statements — Break down the given conditional statement into its component statements: $p$ (the 'if' part) and $q$ (the 'then' part).
2. Formulate the Converse — Construct the converse by writing 'If $q$, then $p
   . Swap the positions of the two components directly without changing their signs.
3. Formulate the Contrapositive — First, find the negations of both statements: $\sim p$ and $\sim q$. Then, construct the statement 'If $\sim q$, then $\sim p
   . Ensure you apply the rules of negation correctly, especially when words like 'all', 'some', or 'not' are involved.

CBSE Board Exam Traps & Tips

1. The 'Only If' Confusion: Remember that 'p only if q' translates to 'If p, then q'. It does NOT mean 'If q, then p'.
2. Logical Equivalence: A conditional statement is always logically equivalent to its contrapositive. However, a statement is NOT necessarily equivalent to its converse. Keep this in mind when proofs require establishing logical equivalence.
3. Double Negatives: When negating a statement that already contains 'no' or 'not', be careful. Negating 'x is not an even number' results in 'x is an even number'.

Practice Questions with Solutions

• Q: Write the contrapositive and converse of the following statement: 'If a number is divisible by 9, then it is divisible by 3.' A: Step 1: Identify the component statements $p$ and $q$. Let $p$: A number is divisible by 9. Let $q$: A number is divisible by 3. Step 2: To find the converse ($q \implies p$): Swap $p$ and $q$. Converse: 'If a number is divisible by 3, then it is divisible by 9.' Step 3: To find the contrapositive ($\sim q \implies \sim p$): Negate both statements and swap them. $\sim q$: A number is not divisible by 3. $\sim p$: A number is not divisible by 9. Contrapositive: 'If a number is not divisible by 3, then it is not divisible by 9.' Final answer: Converse is 'If a number is divisible by 3, then it is divisible by 9.' Contrapositive is 'If a number is not divisible by 3, then it is not divisible by 9.'
• Q: Rewrite the statement 'If you get an A grade, then you worked hard' in five equivalent ways. A: Step 1: Identify $p$ and $q$. $p$: You get an A grade. $q$: You worked hard. Step 2: Use the standard equivalent representations of $p \implies q$: 1. $p$ implies $q$: 'Getting an A grade implies that you worked hard.' 2. $p$ is sufficient for $q$: 'Getting an A grade is sufficient for knowing you worked hard.' 3. $q$ is necessary for $p$: 'Working hard is necessary for getting an A grade.' 4. $p$ only if $q$: 'You get an A grade only if you worked hard.' 5. If $\sim q$, then $\sim p$: 'If you did not work hard, then you did not get an A grade.' Final answer: The five equivalent statements are detailed above.
• Q: Write the contrapositive of the statement: 'If a triangle is equilateral, then it is isosceles.' A: Step 1: Identify component statements. $p$: A triangle is equilateral. $q$: A triangle is isosceles. Step 2: Negate both statements. $\sim p$: A triangle is not equilateral. $\sim q$: A triangle is not isosceles. Step 3: Combine as 'If $\sim q$, then $\sim p
  . Contrapositive: 'If a triangle is not isosceles, then it is not equilateral.' Final answer: 'If a triangle is not isosceles, then it is not equilateral.'
• Q: Check if the converse of the statement is true: 'If x is an even integer, then x is divisible by 2.' A: Step 1: Write down the converse statement. Original: 'If x is an even integer ($p$), then x is divisible by 2 ($q$).' Converse ($q \implies p$): 'If x is divisible by 2, then x is an even integer.' Step 2: Check the validity of the converse. An integer divisible by 2 is, by definition, an even integer. Therefore, the converse statement is true. Final answer: The converse statement is 'If x is divisible by 2, then x is an even integer', which is true.

Frequently Asked Questions