Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these,...

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these, list all ordered pairs in the relation.

Expert Solution

Colby Messinger answered 2 years 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 S1 and S2 be any two equivalence relations on some set A, where A ≠...

Let S1 and S2 be any two equivalence relations on some set A, where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A. Prove or disprove (all three): The relation S defined by S=S1∪S2 is (a) reflexive (b) symmetric (c) transitive

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)

Let A = {1,2,3}. In each part, give an example of the requested function and justify...

Let A = {1,2,3}. In each part, give an example of the requested function and justify that your example has the required properties, or explain why no such function exists. (a) A function f : A → A that does not have an inverse. (b) A function f : A → A×A that is surjective (i.e., onto). (c) A function f : P(A) → A such that for all X ∈P(A), f(X) ∈ X. (d) A function f : A...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

Let A = Σ*, and let R be the relation "shorter than." Determine whether or not...

Let A = Σ*, and let R be the relation "shorter than." Determine whether or not the given relation R, on the set A, is reflexive, symmetric, antisymmetric, or transitive.

For each of the following relations, determine if f is • a function, • surjective, or...

For each of the following relations, determine if f is • a function, • surjective, or • injective. Conclude by stating if the relation represents a bijective function. For each point, state your reasoning in proper sentences. a) f = {(a, b) ∈ N 2 × N | a ∈ N 2 , a = (a1, a2), b, a1, a2 ∈ N, b = a1a2} b) f = {(x, y) ∈ S 2 | y = x 2}, where S...

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

A)Let S = {1,2,3,...,18,19,20} be the universal set. Let sets A and B be subsets of...

A)Let S = {1,2,3,...,18,19,20} be the universal set. Let sets A and B be subsets of S, where: Set A={3,4,9,10,11,13,18}A={3,4,9,10,11,13,18} Set B={1,2,4,6,7,10,11,12,15,16,18}B={1,2,4,6,7,10,11,12,15,16,18} LIST the elements in Set A and Set B: { } LIST the elements in Set A or Set B: { } B)A ball is drawn randomly from a jar that contains 4 red balls, 5 white balls, and 9 yellow balls. Find the probability of the given event. Write your answers as reduced fractions or whole numbers. (a) PP(A...

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