In order to prove that a statement is false, you must write out its negationand then prove that the negation is true. Answer counting questions with a detailed recipeand simplify your answer to a number.Recall that for any finite set S, |S| denotes the number of elements in S.Let A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {1, 3, 5, 7, 9}. Let R be the relation on P(A),the power set of A, defined by:For all X, Y ∈ P(A), XRY ⇐⇒ |X ∪ B| = |Y ∪ B|.(a) Prove that R is an equivalence relation on P(A)(b) How many equivalence classes are there? Explain.(c) How many elements of [∅], the equivalence class of ∅, are there? Explain.(d) Let C = {0, 1, 2, 3, 4}.How many elements of [C], the equivalence class of C, are there? Explain.
In order to prove that a statement is false, you must write out its negation
and then prove that the negation is true. Answer counting questions with a detailed recipe
and simplify your answer to a number.
Recall that for any finite set S, |S| denotes the number of elements in S.
Let A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {1, 3, 5, 7, 9}. Let R be the relation on P(A),
the power set of A, defined by:
For all X, Y ∈ P(A), XRY ⇐⇒ |X ∪ B| = |Y ∪ B|.
(a) Prove that R is an equivalence relation on P(A)
(b) How many equivalence classes are there? Explain.
(c) How many elements of [∅], the equivalence class of ∅, are there? Explain.
(d) Let C = {0, 1, 2, 3, 4}.
How many elements of [C], the equivalence class of C, are there? Explain.
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