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!

Expert Solution

Colby Messinger answered 1 year ago

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 = {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.

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 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 x be a set and let R be a relation on x such x is...

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

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.

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)

For the following relations on S = {0,1,2,3}, specify which of the properties (R), (AR), (AS),...

For the following relations on S = {0,1,2,3}, specify which of the properties (R), (AR), (AS), and (T) the relations satisfy. (R) = reflexive (AR) = anti reflexive (AS) = anti symmetric (T) = transitive b.) (m,n) in the domain of R2 if m - n is even c.) (m,n) in the domain of R3 if m less than or equal to n d.) (m,n) in the domain of R4 if m + n is less than or equal to...

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s...

Let L = {x = a^rb^s | r + s = 1mod2}, I.e, r + s is an odd number. Is L regular or not? Give a proof that your answer is correct.

Question