Let A = {a, b, c, d, e} and define an equivalence relation R C A × A on the set A as follows: R = {(a, a), (b, b), (c, c), (d, d), (e, e), (a, c), (c, a), (b, d), (d, b)}.

Q: Fix a set X, and let S be a collection of equivalence relations on X. Is s U an equivalence relation…

A:

Q: 2. Consider the relation on A = {(1,0), (-1,0), (1,3), (2, 4), (-4,-8), (3,9), (3,6)}, defined by:…

A:

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 A be the set of all triangles defined by R = {(A1, A2) : A1 =A2). Show that R is an equivalence…

A:

Q: Let R be the equivalence relation on the set A ={ a, b, c, d } induced by the partition ( {a,b,c};…

A: Given A=a, b, c ,dpartitiona,b,c; d

Q: Given a non empty set X, consider P(X) which is the set of all subsets of X.      Define the…

A: The relation is defined as follows.

Q: Let A = {a,b, c, d, e, ƒ}, and let G and H be the following equivalence relations in A: G = IA U…

A:

Q: Determine whether the following relation is an equivalence relation? R= {(a, a), (a, c), (c, a), (b,…

A:

Q: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b) : b = a + 1} is…

A: Given, relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1}.

Q: Let Z be the set of integers. Consider R = {(r, y) e Z x Z: 7 divides x-y} CZ x Z. Show that R is an…

A:

Q: Let Z be the set of integers. Consider R = {(x, y) e Z × Z: 7 divides x-y}C Z × Z. Show that R is an…

A:

Q: 4. Let R be a relation defined on the set of integers by a R b if a +b = 0 or a – b = 0. Determine…

A:

Q: Let R be the relation on set {a,b,c,d} containing ordered pairs (a,b),(b,b),(b,c)(c,a)(c,c) and…

A: Given set is A = {a, b, c, d} and the relation R on set A is R = {(a, b), (b, b), (b, c), (c, a),…

Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

A:

Q: Give an example of two equivalence relations R and S on

A: Consider that R and S are equivalence relations on A=1, 2, 3.

Q: Let Z be the set of integers. Consider R = {(x, y) e Z x Z: 7 divides x-y} S Z x Z. Show that R is…

A:

Q: Let S be the relation on the set R^2 defined by (x, y) S (w, z) if |x| + |y| = |w| + |z|. Prove or…

A: Definition of an Equivalence Relations: A relation R on a set A is an equivalence relation if…

Q: Let ~ CX *X be an equivalence relation on a set X. For x EX define x := := {x |x E X} is a partition…

A: This is a question from Relation.

Q: Enter an example of an equivalence relation R on the set X = {0, 1, 2, 4, 8) such that all of the…

A: We need to find an example of an equivalence relation on set X=0,1,2,4,8 such that: 1,4∈R 8,4∉R…

Q: Let A = {a, b, c, d} and R = {(a, a), (b, a), (b, b), (c, c), (d, c), (d, d)}. Determine whether the…

A:

Q: Consider the relation R on the set {a,b,c,d,e,f} which is defined as follows: R={(a,a), (a,b),…

A:

Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…

A: According to the given information, Consider a relation R on R2 as:

Q: Let R = { ( a,b ) | a,b ∈ Q and a − b ∈ Z } . Prove that R is an equivalence relation on Q

A:

Q: Subject: Set Theory

A:

Q: Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists CEP…

A:

Q: 9.29. Let R be an equivalence relation defined on a set A containing the elements a, b, c and d.…

A:

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

A:

Q: If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence…

A: Given, R1 and R2 are equivalence relations in a set A. So, both R1 and R2 are Reflexive, Symmetric…

Q: Show that the relation R defined on the set of integers ℤ by (a,b) Î R if a – b = 6k for some k Î ℤ…

A:

Q: Let R be the relation over the set of all people where aRb iff a is the same age as b. Determine and…

A:

Q: Let RR be a binary relation on A={a,b,c}A={a,b,c} given by…

A: We check relationship is reflexive or not

Q: (a, b, c, d) and define a relation R on A as follows: R = ((a, a), (b, b), (a, b), (c, c), (b, a),…

A:

Q: 2. Let R be a relation defined on the set of integers Z where a R b if a +b is even. Determine…

A:

Q: ) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: Let R be a relation defined on Z as follows: (a,b) is an element of R if and only if the square of a…

A:

Q: let R be an equivalence relation and let [x] be the equivalence class of x, prove x belong to [x]

A:

Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

A:

Q: Let A be a set and R be a reflexive relation on A. Prove that R is an equivalence relation if and…

A:

Q: '17. Prove or disprove: Let R be an equivalence relation defined on a set A. For any x, y E A,…

A: Given result is: '' Let R be an equivalence relation defined on a set A. For any x,y∈A, either x∩y=∅…

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

A:

Q: . Let R be a relation defined on the set A = N x N where (a, b) R (c, d) if a³ s an equivalence…

A: Introduction: Any two elements are said to be equivalent if they have an equivalence relationship. '…

Q: Let A={(m,n)∣m,n∈N2 and gcd(m,n)=1}, and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc.…

A:

Q: Consider the relation R on the set Z defined by a R b if and only if 3|(a+2b) for a, b ∈ Z. Show…

A:

Q: (5) Let A={-2,-1,0,1,2}. R is an equivalence relation defined as: for all x,y E A, xRy 2| (x-y?).…

A:

Q: (1) Let A-(a,b,c,d), define a relation R on set A as follows: R-((a,a). (a,b), (b,a), (b,b), (c,c),…

A: Reflexive relation:  A relation R on set X is said to be reflexive if for all elements x∈X we have…

Q: Let X = {1, 2, 3, 4, 5} and Y = {3, 4}. We define the relation R on the set P (X) by the formula (A…

A: Here, I have shown that the given relation is an equivalence relation and found out the equivalence…

Q: Let R and S be equivalence relations on a set A; recall that by definition R, S ⊆ A × A. Prove that…

A:

Q: Let R be a relation on {a, b, c, d}) defined as R = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c),…

A: Concept:

Q: Show that the relation R defined on the set of integers ℤ by (a,b) ∈ R if a – b = 6k for some k ∈ ℤ…

A:

Q: 2. Let R be the relation on Z→Z such that ((a,b),(c,d)) = R a+d =b+c. Show that R is an equivalence…

A: