If A={4,6,10,12} and R is a relation defined on A as "two elements are related if they have exac

step 1 Hello students in this question we are given a set a equals to 4, 6, 10 and 12 okay we are given

If A={4,6,10,12} and R is a relation defined on A as "two...

If $\mathrm{A}=\{4,6,10,12\}$ and $\mathrm{R}$ is a relation defined on $\mathrm{A}$ as "two elements are related if they have exactly one common factor other than $1 "$ ". Then the relation $\mathrm{R}$ is
(1) anti symmetric.
(2) only transitive.
(3) only symmetric.
(4) equivalence.

Key Concepts

Equivalence Relation

An equivalence relation is one that is reflexive, symmetric, and transitive. Such a relation partitions a set into equivalence classes where each element is considered equivalent under the relation, creating groups of elements that share a common relationship.

Binary Relation

A binary relation on a set is a collection of ordered pairs of elements from that set. It defines a relationship between elements where a specific condition is satisfied, usually determined by some rule or property that the elements share.

Symmetric Relation

A relation is symmetric if whenever an element x is related to an element y, then y is also related to x. This property ensures that the relationship is mutually shared between the pair of elements, reflecting a balance in the connection.

Transitive Relation

A relation is transitive if whenever an element x is related to an element y and element y is related to an element z, then x must also be related to z. This property is crucial in establishing a chain of relationships where the connection between elements extends through intermediate relationships.

Antisymmetric Relation

A relation is antisymmetric if for any two distinct elements x and y, whenever x is related to y and y is related to x, it must follow that x equals y. This restricts mutual relationships among different elements, ensuring that only identical elements can have a two-way relationship.

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