Sage-Tips

Is xy ≥ 0 an equivalence relation?

(iv) An integer number is greater than or equal to 1 if and only if it is positive. Thus the conditions xy ≥ 1 and xy > 0 are equivalent.

Is xy an equivalence relation?

An equivalence relation on a set S, is a relation on S which is reflexive, symmetric and transitive. Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. The parity relation is an equivalence relation.

Is X Y 2k an equivalence relation?

The relation A defined on R by xAy⟺∃k∈Z, x−y=2k is an equivalence relation. Then xAx is relation iff x−x=2k. Hence A is reflexive Since k = 0 and k∈Z.

Which of the following is not an equivalence relation on the set of integers?

Explanation: x y, x ≤ y, R is reflexive and transitive if R is a relation defined by xRy. It is not, however, symmetric. As a result, R isn’t an equivalence relationship.

Is the following binary relation R on Z An equivalence relation justify your answer?

Answer: Yes, R is an equivalence relation. To prove this, I need to show that R is reflexive, symmetric, and transitive. Reflexive: Let x ∈ Z. Then x2 ≡ x2 (mod 4), so xRx.

Is xy >= 0 transitive?

Transitive: No. Let x = z = 1 and y = 0. Then, xy =0= yz but xz = 1 = 0.

Which of the following relation is equivalence relation?

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation is equal to is the canonical example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes.

What is relation equivalence relation?

Definition 1. An equivalence relation is a relationship on a set, generally denoted by “∼”, that is reflexive, symmetric, and transitive for everything in the set. Example: The relation “is equal to”, denoted “=”, is an equivalence relation on the set of real numbers since for any x, y, z ∈ R: 1.

Which of the following relation is not an equivalent relation?

R4 on Z defined by aR4 b ⇔ a-b is an even integer for all a,b∈Z. We observe that (1,1/2)∈R2 and (1/2,-1)∈R2 but (1,-1)∈R2. So, R2 is not a transitive relation. Hence, it is not an equivalence relation.

Is r1 Union R2 equivalence relation?

Thus R1 ∩ R2 is reflexive, symmetric and also transitive. Thus R1 ∩ R2 is an equivalence relation.