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+12 votes

The binary relation $S= \phi$ (empty set) on a set A = {1,2,3} 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 (68.8k points)
retagged by | 1.3k 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.

2 Answers

+20 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 (49.1k 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.
+4 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 (291 points)

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