Which one is an equivalence relation in the set of integers?
The relation a ≡ b(mod m), is an equivalence relation on the set of integers. Proof. Reflexive. If a is an arbitrary integer, then a − a = 0 = 0 · m.
Which of the following relation on the set of integers is not an equivalence relation?
Solution : If R is a relation defined by `x R y : if x le y`, then R is reflexive and transitive. But, it is not symmetric. Here, R is not an equivalence relation.
What are the three conditions for equivalence relation?
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. The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”.
How do you know if its equivalence relation or not?
To prove an equivalence relation, you must show reflexivity, symmetry, and transitivity, so using our example above, we can say:
- Reflexivity: Since a – a = 0 and 0 is an integer, this shows that (a, a) is in the relation; thus, proving R is reflexive.
- Symmetry: If a – b is an integer, then b – a is also an integer.
What is a equivalence relation in sets?
An equivalence relation is a relationship on a set, generally denoted by “∼”, that is reflexive, symmetric, and transitive for everything in the set. 1. (Reflexivity) a ∼ a, 2. (Symmetry) if a ∼ b then b ∼ a, 3. (Transitivity) if a ∼ b and b ∼ c then a ∼ c.
What are the properties of equivalence relation?
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.
Which of the following is not an equivalence relation aRb?
Solution : if `aRb Leftrightarrow a lt b ` As, `a lt b` doesn’t imply `b lt a`, `therefore` relation is not symmetric. thus neither it is a equivalence relation.
What are the properties of an equivalence relation?
Is null set an equivalence relation?
Let S=∅, that is, the empty set. Let R⊆S×S be a relation on S. Then R is the null relation and is an equivalence relation.
Which is not an equivalence relation?
Non-example: The relation “is less than or equal to”, denoted “≤”, is NOT an equivalence relation on the set of real numbers. For any x, y, z ∈ R, “≤” is reflexive and transitive but NOT necessarily symmetric. 1. (Reflexivity) Of course x ≤ x is true since x = x.
What is an equivalence relation on a non empty set a?
In order that a relation R defined on a non – empty set A is an equivalence relation, it is sufficient, if R. A relation is said to be an equivalence relation if it is reflexive, symmetric and transitive in nature.
Can an equivalence relation be empty set?
An equivalence relation on a non empty set can’t be empty, because it’s reflexive. So, for any a∈A, you have (a,a)∈R.
Can an equivalence relation be empty?
Therefore, no equivalence class is empty and the union of all equivalence classes is the whole set A. So the only thing that remains to be shown is that two distinct equivalence classes don’t overlap.