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Question

Let X be a family of sets and R be a relation on X defined by A is disjoint from B. Then R is

A
Reflexive
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B
Symmetric
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C
Anti-symmetric
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D
Transitive
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Solution

The correct option is A Symmetric
Let A be a non-empty set in X.
AA=Aϕ
Hence, A is not related to A.
Hence, R is not reflexive.
Let A and B be the disjoint sets in X
Let ARB
AB=ϕ
or, BA=ϕ
BRA
Hence, R is symmetric.
Let, A,B,CX

Let ARB and BRC

AB=ϕ,BC=ϕ
It can happen that A and C may have a common element.
W can show by example A={1,2,3,4},B={5,6,7},C={4,8}
Here, AB=ϕ,BC=ϕ but AC={4}
Hence, R is not transitive.

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