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Which of the following is an equivalence relation on the set of all functions from Z to Z ?
  • a)
    { (f, g) | f (x) − g (x) = 1 x ϵ  Z }
  • b)
    { (f, g) | f (0) = g (0) or f (1) = g (1) }
  • c)
    { (f, g) | f (0) = g (1) and f (1) = g (0) }
  • d)
    { (f, g) | f (x) − g (x) = k for some k ϵ  Z }
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Which of the following is an equivalence relation on the set of all fu...
(1) { (f, g) | f (x) − g (x) = 1 x ϵ  Z }
It is not reflexive. As f(x) – f(x) = 0 it is not 1. It is also not transitive. So, it cannot be an equivalence relation.
(2) {(f, g) | f (0) = g (0) or f (1) = g (1) }
This relation is not transitive. Suppose f(x) = 0, g(x) = x and h(x) = 1. Here f is not related to h. Only we have a relation given between f and g, g and h. but not between f and h. So, it cannot be an equivalence relation.
(3) {(f, g) | f (0) = g (1) and f (1) = g (0) }
It is not always true f(0) = f(1) for reflexive case. So, it is not reflexive. Hence, no equivalence relation.
(4) { (f, g) | f (x) − g (x) = k for some k ϵ  Z }
It is reflexive relation, consider constant as 0. It is also symmetric because the difference will be equal to a constant value. It is also transitive. So, it is an equivalence relation.
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Which of the following is an equivalence relation on the set of all functions from Z to Z ?a){ (f, g) | f (x) − g (x) = 1 x Z }b){ (f, g) | f (0) = g (0) or f (1) = g (1) }c){ (f, g) | f (0) = g (1) and f (1) = g (0) }d){ (f, g) | f (x) − g (x) = k for some k Z }Correct answer is option 'D'. Can you explain this answer?
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