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+16 votes

The binary relation $S= \phi \text{(empty set)}$ on a set $A = \left \{ 1,2,3 \right \}$ is 

  1. Neither reflexive nor symmetric
  2. Symmetric and reflexive
  3. Transitive and reflexive
  4. Transitive and symmetric
asked in Set Theory & Algebra by Veteran (69k points)
edited by | 1.4k views
It's not reflexive. but is it transitive and symmetric? Please reply.
if set A is also empty set then in that case , set S will become Reflexive.

3 Answers

+21 votes
Best answer

answer = option D

S=ϕ (empty set) on a set A = {1,2,3} is Irreflexive, Symmetric, Anti Symmetric, Asymmetric, Transitive..
but it is not Reflexive

answered by Veteran (54k points)
edited by
why it is not reflexive ?
what is meaning of S= empty set on set A={1,2,3}  ??

learncp A relation 'R' on set A said to be reflexive if  ( x R x ) ∀x ∈ A . It means all diagonal element should be present here .But here no (x,x) type relation is present in given empty set. thats why it is not reflexive.

got it ...

@leaencp @LeenSharma

why it is not reflexive ??

Because ... S does not contain (1,1), (2,2) and (3,3) ordered pair.
yes,Right . All (1,1) (2,2) and (3,3) should be present in the relation .
can anyone explain how it is Irreflexive, Symmetric, Anti Symmetric, Asymmetric, Transitive by giving example.

 Ritesh Pratap Singh this question is the simplest example 

+7 votes
A relation on a set A is by definition a subset R⊆(A×A). Then "a is related to b" means "(a,b)∈R. The empty relation is then just the empty set, so that "a is related to b" is always false.
Hence R= empty set implying it is symmetric,antisymmetric and transitive trivially but not reflexive since by definition any reflexive relation should contain all elements of the form (a,a) for all a in A
answered by (363 points)
0 votes

Relation S is defined as a set in which no element of A is related to any element. 

This means that R is an empty relation.

Empty relation with empty set holds Reflexivity, Transitivity, Symmetric, Anti-Symmetric
This empty relation would match all conditions vacuously because there are no conditions to check(no elements)

Empty relation with non-empty set holds Transitivity, Symmetric, Anti-Symmetric but not reflexivity because set is non-empty and there are conditions to check for reflexive property.

In this particular question set A is non empty hence answer is D) Transitive and symmetric

answered by Boss (9.1k points)

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