Let f(n,k) be the number of equivalence relations with k classes on set with n elements....

Let f(n,k) be the number of equivalence relations with k classes on set with n elements.

a) What is f(2,4)?

b) what is f(4,2)?

c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k * f(n-1,k)

Expert Solution

Colby Messinger answered 1 year ago

9.2 Give 3 examples of equivalence relations and describe the equivalence classes. 9.3 Let R be...

9.2 Give 3 examples of equivalence relations and describe the equivalence classes. 9.3 Let R be an equivalence relation on a set S. Prove that two equivalence classes are either equal or do not intersect. Conclude that S is a disjoint union of all equivalence classes.

Let R and S be equivalence relations on a set X. Which of the following are...

Let R and S be equivalence relations on a set X. Which of the following are necessarily equivalence relations? (1)R ∩ S (2)R \ S . Please show me the proof. Thanks!

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

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Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈ Z,b...

Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈ Z,b ∈ N} (recall that [(a,b)] = {(c,d) ∈Z×N : (a,b) ∼ (c,d)}). We deﬁne an addition and a multiplication on X as follows: [(a,b)] + [(c,d)] = [(ad + bc,bd)] and [(a,b)]·[(c,d)] = [(ac,bd)] Prove that this addition and multiplication is well-deﬁned on X.

Question

Let f(n,k) be the number of equivalence relations with k classes on set with n elements....

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