Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric and antisymmetric.

Q: Let F = {Fα : α ∈ A} be a family of compact subsets of R. Prove that an intersection, α∈A, Fα is…

A: A set F is said to be a compact set if it is closed and bounded both.

Q: Suppose that R is a reflexive and transitive relation on a set A. Define a new relation E on A by…

A:

Q: d. Ris a relation on Z* such that (x, y) e R if and only if there is a positive integer n such that…

A:

Q: Theorem: If R is a fuzzy relation on the set X, then (R-1)-1 = R

A:

Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

Q: PROVE THAT IF A RELATION X ON NON EMPTY SET B IS REFLEXIVE AND TRANSITIVE THEN THE RELATION X…

A:

Q: 3. Let R be the relation defined on Z by aRb if and only if 5 | (4a + b). Prove that R is an…

A: We have given relation on ℤ and defined by  aRb if and only if 5|(4a+b). We can  see here for any…

Q: If a relation R on a set A is antisymmetric, then R is not symmetric.

A: To determine whether the given statement is true or false.

Q: Let R be a relation defined on P(Z) defined by (A, B) E R if and only if AN B + Ø. Prove or disprove…

A:

Q: A = Z*; a R b if and only if |a – b| < 2.

A:

Q: let R be the relation on the set A = {1, 2, 3, 4, 5, 6} defined by a R bif and only if a = 2b.

A:

Q: Consider a relation ~ on the set Z such that x~y if and only if x + y is even. The relation ~ is…

A:

Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

A:

Q: A and B are open subsets of X with Cl(A) = X = Cl(B).

A:

Q: 1. Prove or Disprove: For any set A, there exists a relation R on A such that R is both symmetric…

A:

Q: 1. Define a relation R on R? by stating that (a, b)R(c, d) if and only if a? + b <c? +d². Show that…

A:

Q: (a) x and y are in the same major. (b) x and y have the same last name. (c) x and y are taking a…

A: See the attachment

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

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: Q7. Let R be a relation on the set Z defined by the following rule: for all a, b e Z, a Rb if and…

A:

Q: Let X = {6, 7, 8, 9) and let the relation R on X be given by R = {(x, y): 3 z e N₁ x + z = y}, where…

A:

Q: 4. Let X be any finite set and define the relation R on P(X) by R= {(E,F)||E| = |F\, E, F € P(X)}.

A:

Q: Give an example of a nonempty relation R on A = {1, 2, 3, 4} which is a. symmetric and transitive…

A:

Q: 1. Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric,…

A: solution

Q: Let T be the relation defined by, For all a, b e Z, aTb iff a – b is odd. Is T an equivalence…

A:

Q: Suppose f, g, and h are all mappings of a set A into itself. a. Prove that if g is onto andf 0 g = h…

A: Given that, f,g and h are all mappings of a set A into itself. If g is onto and f∘g=h∘g  To prove…

Q: Define a relation Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x R,…

A: The relation Q on the set R×R is defined as follows. For all ordered pairs w, x and y, z in R×R, w,…

Q: Let R be the relation on the set of integers such that a R b if and only if a = b or a = - b.…

A:

Q: Let A be a nonempty set. Let ~ be a relation on ℘(A)defined by: Letting B and C be arbitrary…

A:

Q: Let R be the relation on the set of real numbers such that x R y if and only if x and y are real…

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 = Z+ x Z+. Define a relation R on A as follows: For all (x, y) and (z, w) in A, (r, y)R(z, w)…

A: We will prove the given statement.

Q: A relation R on a set A is backwards transitive if, and only if, for every r, y, z E A, if rRy and…

A:

Q: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

A:

Q: t R be the relation on the set of ordered pairs of positive integers ch that ((a,b), (c,d)) e R if…

A:

Q: 1. if R and T are anti-symmetric relations on a set A then (R U T) is anti-symmetric relation on А.

A:

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. s R reflexive?…

A: We will check one by one what properties R satisfies and which not.

Q: A space X is locally connected if and only if for every open set U of X, each comj

A: Suppose that X is locally connected. Let U be an open set in X and let C be a component of U. If x…

Q: Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

A:

Q: A relation R is defined on the set, Z, of integers by xRy if and only if x+ y 2 5. R is A reflexive…

A:

Q: Let A be a nonempty set and let P be a partition of A. Define a relation R (corresponding to P) on A…

A: Given, A=1,2,3 We can write the above set as, 1,2,3×1,2,3=1,1,,1,2,1,3,2,1,2,2,2,3,3,1,3,2,3,3

Q: 1. Let - be a relation over Zx (Z\ {0}) defined by (a,b)-(c,d) iff ad = bc. Prove that this is an…

A:

Q: Let R be the relation defined on P({1,.…, 100}) by ... ARB if and only if |AUB| is even. Is R…

A: Relation: - A subset of the productA× B could be a relation R from a non-empty set A to a non-empty…

Q: Let G={(m,n)∣m,n∈N x N and gcd(m,n)=1}, and define the relation q on G according to (a,b) q…

A: A totally ordered set is a set G together with a relation ≤ that satisfies the following conditions.…

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

A: An equivalency relation on a set is a reflexive, symmetric, and transitive relationship for…

Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…

A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…

Q: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…

A: A relation to be an equivalence relation must satisfy the following three properties:  1. Reflexive…

Q: Let R be a relation on the set of all integers such that a Rb if and only if a +b = 0. (a) Is R…

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: