Let A be a non-empty set and R a relation on A. R is called irreflexive if (aa)ER for all aEA. Let z be the set of integers and R be the relation on Z defined as follows: aRb if and only if a(mod 3) = (b+1) (mod 3).Then O Ris an equivalence relation O Ris not an irreflexive relation and not a reflexive relation Ris an irreflexive relation None of these

Q: 3. Let the relation R on X = {1,3,5, 6, 8, 9, 11} be such that xRy means "x – y is a multiple of 3".…

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Q: SULU Let R be a relation over the integers (Z). Prove that R= {(a,b) : a mod 4 = b mod 4} is an…

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Q: Define R on Z+ by aRb if = 5 for some integer k. (a) Prove R is an equivalence relation. (b) What is…

A: The solution is as follows:

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Q: R is the congruence modulo 7 relation on Z: For all m, n E Z, mRn iff 7|(m – n). (i) Is the relation…

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Q: Let S be the set of real numbers. If a, b e S, define a - b if a – b is an integer. Show that ~ is…

A: Definition of the equivalence relation helps to prove the required. A relation is said to be an…

Q: Let O be the relation defined on Z as follows. For every  m, n ∈ Z, m O n  ⇔  m − n  is odd. Which…

A: Given O be the relation defined on Z as follows For every  m, n ∈ Z, m O n  ⇔  m − n  is odd.

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Q: Let the relation - on the natural numbers N be defined as follows: if n is even, then n ~ n + 1; if…

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Q: Let Z be the set of integers and R be the relation on Z defined as: aRb if and only if 1+ab > 0.…

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Q: 3. Let S be the congruent modulo 2 relation on Z, that is m S n 2|(m-n). Show the relation

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Q: Let R be a relation on the set of all non-negative integers defined by aRb if and only if a3 - b3 is…

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Q: 7. Decide for each of the following relations whether or not it is an equivalence relation. Give…

A: A relation is known as equivalence relatioon if it is reflexive, symmetric and transitive.

Q: (1) Let R be the relation defined on Z by aRb if 2a + b = 0 (mod 3). (a) Prove that R is an…

A: (a) The relation between a and b represents that 2a+b is a multiple of 3.

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A: To show R is an equivalence relation, we have to show that R is Reflexive, Symmetric, and…

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Q: 11 Let S be the congruent modulo 5 relation on Z , that is m S n → 5|(m – n) . Show the relation S…

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Q: Consider the set NXN .e the set of ordered pairs of positile integers. Let I be the relection ~ in…

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Q: Let R be the relation on the set of natural numbers given by the following rule: (n,m) ∈R if and…

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Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

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Q: a. The relation x = y if and only if x mod 4 == y mod 4 is an equivalence relation. Use this…

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Q: Let S {3k : k E Z} be the set of integer multiples of 3. Let R be a relation defined on Q by x Ry if…

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Q: 3. Let S be the congruent modulo 2 relation on Z,that is m S n→ 2|(m-n) . Show the relation

A: A relation is called symmetric if for any  if  then we have .

Q: a. Prove that the intersection of two equivalence relations on a nonempty set is an equivalence…

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Q: The relation defined as x ~ yx ≡ y (mod7) for x, y ∈ Z in Z Write the equivalence classes by writing…

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Q: Q1. Let A be a set of nonzero integers and let be the relation on A defined by (a, b) - (c, d)…

A: Solution:-

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A: We need to prove 1) Reflexive 2) symmetric 3) transitive.

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Q: Define an equivalence relation Ron the positive integers A = {1,2, 3, 4,..., 20} by mRnif m = 2*n…

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A: First to prove the relation, R, on set of integers Z is equivalence relation. R is defined as:

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Q: I. Determine whether each relation defined on the set of positive integers is reflexive, symmetric,…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.