CBSE Class 11 Maths: Mathematical Reasoning Ex 14.2 NCERT Solutions

Understanding Statements and Negations

In mathematics, a 'statement' or 'proposition' is a declarative sentence that is either true or false, but not both. For instance, 'The sun rises in the east' is a statement. 'What is your name?' is not, as it's a question.

Exercise 14.2 primarily focuses on two key aspects: negation of statements and understanding conditional statements, including their converse and contrapositive. Negation is the process of forming a new statement that is true whenever the original statement is false, and false whenever the original statement is true. It's often expressed by adding 'not' or phrases like 'it is false that' or 'it is not the case that'.

When negating compound statements (statements formed by combining two or more simple statements using connectives like 'and', 'or'), special rules apply, often derived from De Morgan's laws. For example, the negation of 'p and q' is 'not p or not q'. Similarly, the negation of 'p or q' is 'not p and not q'. We also encounter quantified statements, which involve phrases like 'for all' (universal quantifier) or 'there exists' (existential quantifier). Negating these requires a careful swap of the quantifier and negation of the predicate.

Key Rules for Negating Statements

Negation of 'p and q'

The negation of 'p and q' is 'not p or not q'. For example, if p is 'It is cold' and q is 'It is raining', then 'It is cold and it is raining' has the negation 'It is not cold or it is not raining'.

Negation of 'p or q'

The negation of 'p or q' is 'not p and not q'. For example, 'The number is even or it is prime' has the negation 'The number is not even and it is not prime'.

Negation of 'For all x, P(x) is true'

The negation is 'There exists an x such that P(x) is false'. For example, 'For all numbers x, x > 0' has the negation 'There exists a number x such that x ≤ 0'.

Negation of 'There exists an x such that P(x) is true'

The negation is 'For all x, P(x) is false'. For example, 'There exists a bird that can fly underwater' has the negation 'For all birds, they cannot fly underwater'.

Conditional Statements: Converse and Contrapositive

A conditional statement, often written as 'If p, then q' (symbolically p → q), asserts that if statement p is true, then statement q must also be true. Here, p is called the hypothesis or antecedent, and q is called the conclusion or consequent. These statements are fundamental to mathematical proofs.

From a conditional statement 'If p, then q', we can form two other related statements:

1. Converse: The converse of 'If p, then q' is 'If q, then p'. It's formed by swapping the hypothesis and the conclusion. Important: The converse is not logically equivalent to the original statement. Just because a statement is true, its converse might be false. For example, 'If a number is divisible by 4, then it is divisible by 2' is true. Its converse, 'If a number is divisible by 2, then it is divisible by 4', is false (e.g., 6 is divisible by 2 but not by 4).

1. Contrapositive: The contrapositive of 'If p, then q' is 'If not q, then not p'. It's formed by negating both the hypothesis and the conclusion, and then swapping them. Crucially, a conditional statement and its contrapositive are logically equivalent. This means they always have the same truth value. If one is true, the other is true; if one is false, the other is false. This equivalence is incredibly useful in proofs (e.g., proof by contrapositive).

Understanding these relationships is key to analyzing and constructing valid arguments in mathematics.

Worked Examples from Ex 14.2

• Example 1: Negating a Compound Statement Question: Write the negation of the statement: "All integers are positive or negative." Solution: Step 1: Identify the components and connective. Let p be "All integers are positive" and q be "All integers are negative." The statement is of the form "p or q" (though the phrasing implies "Every integer is either positive or negative"). More accurately, let's break it down as 'x is positive' or 'x is negative' for any integer x. Step 2: Rephrase the original statement for clarity if needed. The original statement implies "For every integer x, x is positive or x is negative." Step 3: Apply the negation rules. The negation of 'P or Q' is 'not P and not Q'. The negation of 'For every x, P(x)' is 'There exists an x such that not P(x)'. Step 4: Combine. The negation becomes "There exists an integer x such that x is not positive AND x is not negative." Step 5: Simplify. An integer that is neither positive nor negative is zero. So, the negation is "There exists an integer that is neither positive nor negative," or simply, "Zero is an integer that is neither positive nor negative." Final Answer: "Zero is an integer that is neither positive nor negative."
• Example 2: Finding Converse and Contrapositive Question: Write the converse and contrapositive of the statement: "If a number is odd, then its square is odd." Solution: Step 1: Identify the hypothesis (p) and conclusion (q). p: "A number is odd." q: "Its square is odd." Step 2: Form the Converse. Converse is "If q, then p." Substitute p and q: "If its square is odd, then a number is odd." Final Answer (Converse): "If the square of a number is odd, then the number is odd." Step 3: Form the Contrapositive. Contrapositive is "If not q, then not p." Find 'not q': "Its square is not odd" (or "Its square is even"). Find 'not p': "A number is not odd" (or "A number is even"). Substitute 'not q' and 'not p': "If its square is even, then the number is even." Final Answer (Contrapositive): "If the square of a number is even, then the number is even."
• Example 3: Negating a Quantified Statement Question: Write the negation of the statement: "There exists a triangle which is equilateral and right-angled." Solution: Step 1: Identify the quantifier and the predicate. Quantifier: "There exists a triangle." Predicate P(x): "x is equilateral and right-angled." Step 2: Apply the negation rule for 'There exists'. The negation of 'There exists an x such that P(x)' is 'For all x, not P(x)'. Step 3: Find the negation of the predicate P(x). P(x) is "x is equilateral and x is right-angled." Using De Morgan's Law for 'and', negation of P(x) is "x is not equilateral OR x is not right-angled." Step 4: Combine the negated quantifier and predicate. Negation: "For all triangles, it is not equilateral or it is not right-angled." Final Answer: "For every triangle, it is either not equilateral or it is not right-angled." (or, more simply, "No triangle is both equilateral and right-angled.")

Exam Tips for Mathematical Reasoning

To ace questions on mathematical reasoning, especially from Exercise 14.2, keep these tips in mind:

• Read Carefully: Always read the statement multiple times to fully grasp its meaning before attempting any operation (negation, converse, contrapositive).
• Identify Connectives and Quantifiers: Clearly distinguish between 'and', 'or', 'if-then', 'for all', and 'there exists'. These dictate which negation rules to apply.
• Negation of Compound Statements: Remember De Morgan's Laws: 'not (p and q)' is equivalent to 'not p or not q', and 'not (p or q)' is equivalent to 'not p and not q'. Don't forget to negate both parts of the compound statement and flip the connective.
• Negation of Quantifiers: 'For all' becomes 'There exists (such that not)' and 'There exists' becomes 'For all (such that not)'. This is a common point of error.
• Converse vs. Contrapositive: These are frequently confused. The converse swaps the hypothesis and conclusion. The contrapositive swaps AND negates both. Remember that a statement and its contrapositive are logically equivalent, while a statement and its converse are not necessarily.

Practice Questions with Solutions

• Q: Write the negation of the statement: "The sum of two even numbers is always even and positive." A: Step 1: Identify the components. Let p be 'The sum of two even numbers is always even' and q be 'The sum of two even numbers is always positive'. The statement is 'p and q'. Step 2: Apply the negation rule for 'p and q', which is 'not p or not q'. Step 3: Form 'not p': 'The sum of two even numbers is not always even' (or 'There exists a pair of even numbers whose sum is odd'). Step 4: Form 'not q': 'The sum of two even numbers is not always positive' (or 'There exists a pair of even numbers whose sum is not positive/is zero or negative'). Step 5: Combine 'not p' and 'not q' with 'or'. Final answer: "The sum of two even numbers is not always even or the sum of two even numbers is not always positive."
• Q: Write the converse and contrapositive of the statement: "If a number is divisible by 6, then it is divisible by 3." A: Step 1: Identify hypothesis (p) and conclusion (q). p: "A number is divisible by 6." q: "It is divisible by 3." Step 2: Form the Converse ("If q, then p"). Converse: "If a number is divisible by 3, then it is divisible by 6." Step 3: Form the Contrapositive ("If not q, then not p"). Not q: "A number is not divisible by 3." Not p: "It is not divisible by 6." Contrapositive: "If a number is not divisible by 3, then it is not divisible by 6." Final answer: Converse: "If a number is divisible by 3, then it is divisible by 6." Contrapositive: "If a number is not divisible by 3, then it is not divisible by 6."
• Q: Negate the statement: "There is a student in the class who loves both Math and Physics." A: Step 1: Identify the quantifier and predicate. Quantifier: "There is a student in the class." Predicate P(x): "x loves both Math and Physics." This is a compound statement: 'x loves Math AND x loves Physics'. Step 2: Apply negation rule for 'There exists'. Negation of 'There exists x, P(x)' is 'For all x, not P(x)'. Step 3: Negate the predicate P(x). Negation of 'x loves Math AND x loves Physics' is 'x does not love Math OR x does not love Physics'. Step 4: Combine. Final answer: "For every student in the class, they do not love Math or they do not love Physics." (Alternatively: "No student in the class loves both Math and Physics.")
• Q: Rewrite the statement "You can take the exam only if you have paid the fees" in the "if-then" form. A: Step 1: Understand the meaning of "only if". "A only if B" means "If A, then B". This implies that B is a necessary condition for A. Step 2: Identify A and B. A: "You can take the exam." B: "You have paid the fees." Step 3: Form the "if-then" statement. Final answer: "If you can take the exam, then you have paid the fees."
• Q: Write the negation of the statement: "All rational numbers are real numbers." A: Step 1: Identify the quantifier and predicate. Quantifier: "All rational numbers" (equivalent to "For every rational number x..."). Predicate P(x): "x is a real number." Step 2: Apply the negation rule for 'For all'. Negation of 'For all x, P(x)' is 'There exists an x such that not P(x)'. Step 3: Negate the predicate P(x). 'not P(x)' is "x is not a real number." Step 4: Combine. Final answer: "There exists a rational number that is not a real number."

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