Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_relations

Class 12 Maths · Relations And Functions · Very Long Answer #182

On A={0,1,…,12}, prove both R₁={(a,b):4 divides a−b} and R₂={(a,b):a=b} are equivalence relations. Find the class of 1 for each.

✅ Step-by-Step Solution

Final Answer: [1]R₁={1,5,9}; [1]R₂={1}.

For R₁, the diagonal difference a−a=0 is divisible by 4, establishing reflexivity.
If a−b=4k, then b−a=4(−k), proving symmetry; if additionally b−c=4l, then a−c=4(k+l), proving transitivity.
The elements of A congruent to 1 modulo 4 are 1,5,9, so [1]R₁={1,5,9}.
For R₂, equality is reflexive (a=a), symmetric (a=b implies b=a), and transitive (a=b,b=c imply a=c); only 1 equals 1.
Both are equivalence relations, with [1]R₁={1,5,9} and [1]R₂={1}.

[1 mark] For R₁, the diagonal difference a−a=0 is divisible by 4, establishing reflexivity.
[1 mark] If a−b=4k, then b−a=4(−k), proving symmetry; if additionally b−c=4l, then a−c=4(k+l), proving transitivity.
[1 mark] The elements of A congruent to 1 modulo 4 are 1,5,9, so [1]R₁={1,5,9}.
[1 mark] For R₂, equality is reflexive (a=a), symmetric (a=b implies b=a), and transitive (a=b,b=c imply a=c); only 1 equals 1.
[1 mark] Both are equivalence relations, with [1]R₁={1,5,9} and [1]R₂={1}.

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