Table of Contents
- 1 How do you prove a relation is reflexive symmetric or transitive?
- 2 Which of the following is an equivalence relation on R for AB ∈ Z?
- 3 What type of relation is less than?
- 4 How do you know if R is transitive?
- 5 How do you prove that a relation is an equivalence relation?
- 6 Can We say every empty relation is an equivalence relation?
How do you prove a relation is reflexive symmetric or transitive?
What is reflexive, symmetric, transitive relation?
- Reflexive. Relation is reflexive. If (a, a) ∈ R for every a ∈ A.
- Symmetric. Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R.
- Transitive. Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R. If relation is reflexive, symmetric and transitive,
Which of the following is an equivalence relation on R for AB ∈ Z?
Which of the following is an equivalence relation on R, for a, b ∈ Z? Explanation: Let a ∈ R, then a−a = 0 and 0 ∈ Z, so it is reflexive. To see that a-b ∈ Z is symmetric, then a−b ∈ Z -> say, a−b = m, where m ∈ Z ⇒ b−a = −(a−b)=−m and −m ∈ Z. Thus, a-b is symmetric.
Which of the relation R on the set of real numbers is an equivalence relation?
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. (Reflexivity) x = x, 2. (Symmetry) if x = y then y = x, 3.
Is the relation An 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 type of relation is less than?
xRz, The relation is transitive.
How do you know if R is transitive?
Transitive: A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A. If there is a path from one vertex to another, there is an edge from the vertex to another.
Is R reflexive symmetric or transitive?
R is reflexive if for all x A, xRx. R is symmetric if for all x,y A, if xRy, then yRx. R is transitive if for all x,y, z A, if xRy and yRz, then xRz. R is an equivalence relation if A is nonempty and R is reflexive, symmetric and transitive.
Which of the following relations are equivalence relations?
Equivalence relations are relations that have the following properties: They are reflexive: A is related to A. They are symmetric: if A is related to B, then B is related to A. They are transitive: if A is related to B and B is related to C then A is related to C.
How do you prove that a relation is an equivalence relation?
Let a − b = c, where c ∈ Q, then b − a = − c. Well if c ∈ Q, then ( − c) ∈ Q as well. So ( b − a) ∈ Q as well. Consider d + e. Since rational numbers are closed under addition, then ( d + e) ∈ Q Since the relation is reflexive, symmetric, and transitive, then the relation is an equivalence relation.
Can We say every empty relation is an equivalence relation?
We can say that the empty relation on the empty set is considered as an equivalence relation. But, the empty relation on the non-empty set is not considered as an equivalence relation. Can we say every relation is a function? No, every relation is not considered as a function, but every function is considered as a relation.
How do you know if a relation is reflexive?
A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive. Reflexive: A relation is said to be reflexive, if (a, a) ∈ R, for every a ∈ A.