On A = {1,2,3,4}, let R = {(a,b) : a^2 = b^2}. What type of relation is R?
Answer and explanation
Correct answer: Equivalence relation
Because every element of A is positive, the equation a^2 = b^2 implies a = b; the alternative a = −b cannot occur within this set for two positive elements. Hence R is exactly {(1,1),(2,2),(3,3),(4,4)}, the identity relation on A. It is reflexive because every (a,a) is included, symmetric because (a,b) implies a=b and therefore b=a, and transitive because a=b and b=c imply a=c. A relation that is reflexive, symmetric, and transitive is an equivalence relation, so option A is correct. Option B is incomplete, option C ignores the other properties, and option D is false.
Frequently asked questions
What is the correct answer to this question?
Equivalence relation
Why is this the correct answer?
Because every element of A is positive, the equation a^2 = b^2 implies a = b; the alternative a = −b cannot occur within this set for two positive elements. Hence R is exactly {(1,1),(2,2),(3,3),(4,4)}, the identity relation on A. It is reflexive because every (a,a) is included, symmetric because (a,b) implies a=b and therefore b=a, and transitive because a=b and b=c imply a=c. A relation that is reflexive, symmetric, and transitive is an equivalence relation, so option A is correct. Option B is incomplete, option C ignores the other properties, and option D is false.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Equivalence relation.
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