NCERT Class 12 · Ch 1 Exercise 1.1 · Q9
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.1 Question 9
On A={x∈ℤ:0≤x≤12}, prove equivalence and find the elements related to 1 for (i) |a−b| a multiple of 4; (ii) a=b.
💡 Concept Hints
- Find all numbers of the form 1+4k that lie in the stated finite set.
- Work through the definitions using the stated domain and codomain; justify a failed universal property with one valid counterexample.
✅ Step-by-Step Solution
Final Answer: (i) {1,5,9}; (ii) {1}.
- (i) The condition is a≡b mod 4. Since 4 divides 0 and divisibility is unchanged by negation, the relation is reflexive and symmetric.
- If 4 divides a−b and b−c, it divides their sum a−c; hence transitive. The elements of A congruent to 1 mod 4 are 1,5,9.
- (ii) Equality is reflexive, symmetric and transitive. An element is equal to 1 precisely when it is 1.
- Therefore the requested classes are {1,5,9} and {1}.
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