Let U = {1, 2, 3} and A = U × U. In each case, show that = is an equivalence on A and find the quotient set A. (a) (a, b) = (a₁, b₁) if a + b = a₁ + b₁. (b) (a,b) = (a₁, b₁) if ab = a₁b₁. (c) (a,b) = (a₁, b₁) if a = a₁. (d) (a,b) = (a₁, b₁) if a − b = a₁ − b₁.

Q: (3) Let S be the equivalence relation on (0, 1, 2, 3} x {0, 1, 2} defined by (a, b)S(c, d) if and…

A: S=0,1,2,3×0,1,2=0,0;0,1;0,21.0;1,1;1,22.0;2,1;2,23.0;3,1;3,2

Q: Let R1 = {(1,1), (2, 2)} and R2 = {(1,1), (1, 2)} be relations on A = {1, 2}. Then R1 U R2 = %3D %3D…

A: R1=1, 1, 2, 2  &  R2=1, 1, 1, 2 are the relations defined on set A=1, 2.

Q: 11. ( a) b) c) ) Consider the relation on A={1,2,3) represented by the matrix Determine if the…

A:

Q: A). C). X y y 7 - 2 - 2 - 1 - 1 - 4 4 - 5 1 1 -4 1 - 2 - 5 1 y 1 y B). D). - 2 - 2 - 1 6. 8. - 1 0.…

A:

Q: Let A = {6, 9} , R relation on A defined R by. %3D R = {(a, b): a +2 = b -1 ,a, b E A}. Write R. %3D

A:

Q: Consider the relation R= {(a, b), (a, c), (c, c), (b, b), (c, b), (b, c)} on the set A = {a,b,c}. Is…

A:

Q: Which of the following is always true about the Cartesian product of sets A and B? A. (A x B) x C =…

A:

Q: In cartesian product, prove that (A-B)XC=(AXC)-(BXC) A,B,C are sets

A:

Q: Exercise 2. Let A, B C R. (a) Prove that AUB = AUB. (b) Prove that ANBCAN B. (c) Find sets A and B…

A: Solution:-

Q: for the following relations on 2 the set A = {1, 2, 3}: R = {(1, 1), (1, 2), (1, 3), (3, 3)},…

A: A relation R is an equivalence relation if it is: i) Reflexive: if aRa for all a in A ii) Symmetric:…

Q: Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C') = (A × B) \ (A × C). Prove that (A x B) n (C…

A:

Q: 4. Let A = (1,2,3} and B= {1,2,3,4). The relations R, = {(1,1),(2,1),(3,2)} and…

A: Since you have posted multiple subparts we solve first three subparts for you To get the solution of…

Q: (2.3) Prove that R is an equivalence relation.

A: Prove R is an equivalnce relation.

Q: Let A = {1, 2, 3, 4, 5, 6} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (2, 3), (3, 2)}.…

A:

Q: Which relation on the set {a, b, c, d} are equivalence relations and contain (i) (b, c) and (c, d)…

A:

Q: R={(0,4), (3,0), (3,1), (4,2)}. Find transitive closer of R. Show diagram. A=Z+ x Z+ show relation R…

A: The question involves finding the transitive closure of a relation R and demonstrating the relation…

Q: Assume that A, B, C and D are sets. Verify that (A x B) U (C x D) ≤ (AUC) x (BUD).

A:

Q: Let X = {1, 2, 3, 4}. Let R = {(1, 3), (2, 1), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (a, b) },…

A:

Q: Let R₁ and R₂ both be equivalence relations. Show by proof or disprove by counterex- ample the…

A:

Q: a) Let A = {4, {$}, ($, [$}}} and B = {1,2}. How many equivalence relations on A?

A:

Q: Define two relations R, and R2 on set positive integers. R1 = {(a,b) | a > b} and R2= {(a,b)| a² = b…

A: Given that two relations R1 and R2 on the set of positive integers.                    R1=(a,b) |…

Q: Let R(a, b, c), S(a, b, c), and T(a, b, c) be three relations. Write one or more Datalog rules that…

A: Given (R∪S)-T The solution for (R∪S) is as follows ∪(a,b,c)←R(a,b,c) OR S(a,b,c)--------(i) The…

Q: Let A = {1,2,3,4} and let R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3),(4,4)} . Show that R is an…

A:

Q: Let A = {a, b, c, d} and let R = {(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c),…

A: Given that   be a relation on the set Reflexive:  For every…

Q: (c) Using the set identities, prove that (A – B) n (A – C) = A – (B U C).

A: To Prove: A-B∩A-C=A-B∪C

Q: Other Math 1) A={1,2,3,4} Relation R on A is R={(1,1),(2,2),(2,3),(3,2),(4,2),(4,4} a. Is R…

A:

Q: Let A, B and C be sets. Use the algebra of sets to verify each statement. 1. A- (B U C) = (A – B) n…

A: Solution

Q: Let A={a,b,c,d} and let R={(a,b)(b,c)(c,d)(d,b)} be a relation on A. Represent the relation using…

A: Given  A = {a, b, c, d} R = { (a, b), (b, c), (c, d), (d, b)}

Q: Let R and S be symmetric relations. Show: R ◦ S symmetric ⇔ R ◦ S = S ◦ R

A:

Q: Using the relation {(1, 2), (1, 4), (3, 3), (4, 1)}, write the relations that have the minimum…

A: Relations are an essential concept in mathematics that describe the connections or associations…

Q: 11. ( a) b) ) Consider the relation on A Determine if the relation is refl Determine if the relation…

A:

Q: 11. ( a) b) c) [1 1 1 ) Consider the relation on A={1,2,3) represented by the matrix 0 10 0 1 1…

A: 11. Consider the relation on A = {1, 2, 3} represented by the matrix 111010011   (a) To determine:…

Q: How many relations are there on the set {a, b}. Show all possible relations on the set {a, b} in…

A: