Examples of Asymmetric Relations

Example: Consider set A = {a, b}

• R = {(a, b), (b, a)} is not asymmetric relation but
• R = {(a, b)} is symmetric relation.

Example: Divisibility Relation:

• Consider the set of positive integers, and define the relation R as “divides.” If a divides b, then b cannot divide a.

R = {(2, 4), (3, 6), (5, 10)}

In this relation, each number on the left divides the corresponding number on the right, but the reverse is not true.

Example: Strict Subset Relation:

• Let’s define a set A and the relation R as “is a strict subset of.” If A is a strict subset of B, then B cannot be a strict subset of A.

R = {{1, 2}, {1, 2, 3}), {a, b}, {a, b, c}}

In this relation, each set on the left is a strict subset of the set on the right, but not vice versa.

Example:

• “Is Less Than” Relation:

Consider the relation “is less than” on the set of real numbers. If a<b, then it is not true that b<a. This is an asymmetric relation.

Example:

• “Is a Proper Subset Of” Relation:

Let’s consider the relation “is a proper subset of” on the set of all sets. If set A is a proper subset of set B, then it is not the case that B is a proper subset of A.

Example:

“Is Predecessor Of” Relation:

• Consider a relation representing the “is predecessor of” relation on the set of integers. If a is the predecessor of b, then it is not true that b is the predecessor of a, except when a=b+1.

Conclusion: Asymmetric Relation

In conclusion, an asymmetric relation is a specific type of binary relation on a set in which the order of elements matters. The key characteristic of an asymmetric relation is that if a pair (a,b) is in the relation, then the pair (b,a) must not be in the relation for any elements a and b from the set. In other words, the relationship is one-directional, ensuring that it does not allow for symmetry.

Read More,

• Set Theory
• Types Of Sets
• Relation and Function
• Types of Functions

A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on.