Which of the following are equivalence relations? If they are not, which of the axioms fails? a) For a, b, k = Z, aRba-bis divisible by k b) Every PhD student has at least two supervisors. Let r, y € A where A is the set of all PhD students, and say that rRy if a shares one supervisor with y.

Q: Let R and S be two equivalence relations on a nonempty set, and T their intersection. What's their…

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Q: Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if…

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Q: Consider the set ℕ× ℕ. We will informally think of the pair (x, y) ∈ ℕ× ℕ as representing the…

A: a) The relation is defined on ℕ×ℕ as follows. x,y~n,m if and only if x+m=n+y. Every x,y~x,y since…

Q: Which of the following are true for all sets A, B, and C. If the statement is true, prove it by…

A: (d) Let, x ∈ (A')'       => x ∉ A'       => x ∈ A      So, (A')' ⊆ A. . . . . . (I)…

Q: Suppose X and Y are subsets of some universal set. De Morgan's laws provide expressions for the…

A: X∪Yc=Xc∩Yc

Q: 4. a) Define the following terms: i) POSETS ii) Equivalence relation ng the relation R and S on

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: 14. y = e, y = 0, x= -1, x = 1; about the x-axis

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Q: a)Let R be a relation on the set Z × Z defined by the following rule: For all (a, b),(c, d) ∈ Z × Z,…

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Q: Given a natural number c ∈ N. On natural numbers, the relation Rc is defined as follows: ∀ a, b ∈ N…

A: "Since you have posted a question with multisubparts, we will solve the first three subparts for…

Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

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Q: Consider the following binary set relation: R= {(1,5), (2, 2), (5, 1)}, on A = {1, 2, 5} a) Compute…

A: Note: The set Bis not given hence w won't be able to solve part (c).  The given relation is…

Q: a) S=R, and a~b a=b or -b b) S=R, and a~b a2+a=b2+b c) S is the set of all people in the world,…

A: As per the company rule, we are supposed to solve the first three sub-parts of a multi-parts…

Q: 8) Let A = {2,3,4, 5}, R is defined on A such that (aRB) if and only if (a2 +B+a) is an even number.…

A: Given A={2,3,4,5} We define a relation R on A by (αRβ) if and only if (α2+β+a) is an even number…

Q: Be Pa binary predicate on a domain Dany. Consider the following sentences in predicate logic: PR =…

A: For (a), Here, we need to show that φR∧φV≡φE. If we can show that φV≡φS∧φT when φR is true for all x…

Q: Let R and T be arbitrary equivalence relations on a universal set S. Verify whether R ∩ T, or R ∪ T…

A: Let R and T be arbitrary equivalence relations on a universal set S. Verify whether R ∩ T, or R ∪…

Q: Can you prove that A ∩ B = B ∩ A for every two sets A and B without using the commutative law of…

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Q: Let F = {e, m, u} and G = {m, a, n} Which of the following statements is true about the cartesian…

A: Let F = {e, m, u} and G = {m, a, n}. The Cartesian product F x G. F X G m a n e (e,m) (e,a)…

Q: Let A = {a, b, c, d, e, f,g, h}, and let G and H be the following equivalence relations in A: %3D G…

A: Given that A = a , b , c , d , e , f , g , h . Also,  G and H are the following equivalence…

Q: 10) Which of the following statements is false? a) fR is reflexive, then RnRep b) ROR' p R is…

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Q: Consider the {(x, y)|x divides to y} the relation defined on the following set S = {3, 4, 6, 7, 11,…

A: As per the rules, we can only solve first 3 parts. Please post the other posts in another question.

Q: a) Let A = {4, 7, 8} and B = {−1, 5}, Find and compare sup(A+B) , inf(A+B), sup(A)+sup(B),sup A -…

A: See the attachment

Q: This question refers to unions and intersections of relations. Since relations are subsets of…

A: The objective of this question is to find the Cartesian product of sets A and B, and the relations R…

Q: Let the universe of elements include these letters: p, q, r, s, t, u, v, w, x, y. The first set A…

A: The universe set contains the elements: p, q, r, s, t, u, v, w, x, y The set A contains the…

Q: 18. How many different equivalence relations are there on the set {a, b, c, d]? O 15 O 12 O 14 18

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Q: Given a natural number c ∈ N. The relation Rc is defined as follows: ∀ a, b ∈ N :  (a, b) ∈ Rc ⇔ (∃…

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Q: 2) Let K be any Boolean algebra. A useful relation y (read as "x precedes y" ) if and only if xy=x.…

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Q: Let A = {p , q, r, s, t, u, v}, and let R be the following equivalence relation on A: R = { (p, p)…

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Q: (а, b, c), В %3D (у), с {a, b, c, x, y, z, 1, 2, 4, 5 }. Find following: 7. Let A = {1,2,5} and U…

A: As per our guidelines, we can answer only three sub-parts in a single request. Kindly re-post the…

Q: Prove and say the name of the following equivalence relation x ~ y ⟺ x - y = 4k, k ∈ Z

A: x ~ y iff x - y = 4k , k in Z Implies that  x is congurent to y mod 4.

Q: Establish the following set theoretic relations: a) AUB=BUA, ANB=BnA (Commutativity) b) AU (BUC) =…

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Q: Suppose that x is a real number and that UCR. Prove that the following two conditions are…

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Q: Let R ⊆ ℝ×ℝ with R={(x,y)|⌈x⌉=⌈y⌉}, that is, x and y round up to the same number. Characterize R in…

A: Given problem is :

Q: 3. (10 points) a) Use an element proof to prove the following: for all sets A and B, A - (ANB) = A –…

A: The detailed solution is as follows below:

Q: a) Define the following terms: i) POSETS ii) Equivalence relation

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Q: Let Z = {1,2, 3, ...,50}. a. How many relations on the set Z are both reflexive and antisymmetric?…

A: Here given set Z=1, 2, 3, ....., 50

Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by  R=1,1,2,2,3,31,2,2,1 And…

Q: >) Recall that the symmetric difference of two sets X and Y, denoted XAY = (X-Y)U(Y-X), is the set…

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Q: 8. Let A = {p, q, r, s, t, u, v}, and let R be the following equivalence relation on A: R = {(p, p),…

A: Equivalence Relation:            "A relation E on a set A is called an equivalence relation if E…

Q: For nonempty sets A, B, and C, |A| <= |B| and |B| <= |C|, then |A| <= |C|.

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Q: Let - be a relation on a set X defined as below. Determine whether is an equivalence relation on X…

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