Short Answer TypeGiven a non-empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows :
For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X) ? Justify your answer.
Long Answer TypeShow that the relation R in the set A = { 1, 2, 3, 4, 5 } given by
R = { (a, b) : | a – b | is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
Short Answer TypeThe smallest equivalence relation R1 containing (1, 2) and (2, 1) is {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}.
Now we are left with only 4 pairs namely (2, 3), (3, 2), (1, 3) and (3, 1).
If we add any one, say (2, 3) to R1. then for symmetry we must add (3, 2) also and now for transitivity we are forced to add (1,3) and (3, 1). Thus, the only equivalence relation bigger than R1 is the universal relation. This shows that the total number of equivalence relations containing (1,2) and (2, 1) is two.
For 
the set of relational numbers, define 
if and only, if a d = b c. Show that R is an equivalence relation on Q.
Long Answer Type
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