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(Please provide detailed solutions ) . Suppose R is a reflexive and symmetric relation on a finite set A. Define a relation S on A by declaring ISy if and only if for some n € N there are elements 21, x2,…, In E A satisfying
tRa1, 71Rx2, x2Rx3,., In-Ron and In Ry.
Show that S is an equivalence relation and R C S. Prove that S is the unique smallest equivalence relation on A that contains R.
2. Consider the partition
Р = 1{0}, 1-1,1), 1-2,2), 1-3,3},…}
of Z. Describe the equivalence relation whose equivalence classes are P.
Consider the function f : Z x Z → Z defined by f(n, m) = 3m – 4n. Is this function injective?
Surjective? Bijective? Explain.
Let f: A → B be a function, and suppose Y B, then prove or disprove the following relation:
1-‘ (f(5-“(Y))) = f-1(Y)
Prove that the set of complex numbers, C, is uncountable.
Let F be the set of all possible functions R → {0,1}. Show that IR| < |F|. (Hint: Think about the relationship between F and subsets of R.)
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