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The binary relation $R = \{(1, 1), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)\}$ on the set $A=\{1, 2, 3, 4\}$ is

  1. reflexive, symmetric and transitive

  2. neither reflexive, nor irreflexive but transitive

  3. irreflexive, symmetric and transitive

  4. irreflexive and antisymmetric

in Set Theory & Algebra by Veteran (52.3k points)
edited by | 1.5k views

3 Answers

+22 votes
Best answer
Not reflexive - $(4,4)$ not present.

Not irreflexive - $(1, 1)$ is present.

Not symmetric - $(2, 1)$ is present but not $(1, 2)$.

Not antisymmetric - $(2, 3)$ and $(3, 2)$ are present.

Not Asymmetric - asymmetry requires both antisymmetry and irreflexivity

It is transitive so the correct option is $B$.
by Veteran (61k points)
edited by
I am not getting 4,4 not present means not reflexive but 1 1 is there so not irreflexive ...plz explain.
a reflexive relation must contain all elements of the form (x,x) and an irreflexive relation should not contain any such pair.
for element (2,4) transistive pair??
y its nt anti symmetric ??
Because it contains 2,3 as well as 3,2
if aRb and bRa ... then a=b .. this is anti symmetric right ??
Yes true.
then (2,3) and (3,2) is in relation... so cant we say that its an anti symmetric relation ??
anti symetric allows only self loop and nothing else.
How can we say that ? See they are symmetric pairs and not allowed in antisymmetric relation. Only diagonal pairs like 3,3 are allowed in antisymmetric relation. So it's not antisymmetric.
so one can conclude that anti symmetric is one kind reflexive relation right ??
reflecixe relations have all the reflexive pair
but antisymetric doesnot need that property.
Not at all. Antisymmetric only means that symmetric pairs shouldn't be present except the diagonal pairs if at all. Even empty relation is antisymmetric but not reflexive.
According to B) neither reflexive, nor irreflexive is fine.

but it is transitive for only one relation (2,3), (3,3) , (3,2).

No other relation is transitive
It is not transitive as (1,1) is present and (2,2) is present but (1,2) is not present
@suchi matkar your example isn't following property of transitive relation which is if (a R b) and (b R c) then (a R c).
0 votes

option b

by Boss (36.9k points)
–3 votes

The correct answer is,(B)neither reflexive, nor irreflexive but transitive

by Loyal (8.1k points)

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