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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

asked in Set Theory & Algebra by Veteran (59.6k points)
edited by | 947 views
+1

This might help ...

3 Answers

+18 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 $B$.
transitive.
answered by Veteran (55.4k points)
edited by
0
I am not getting 4,4 not present means not reflexive but 1 1 is there so not irreflexive ...plz explain.
0
a reflexive relation must contain all elements of the form (x,x) and an irreflexive relation should not contain any such pair.
0
for element (2,4) transistive pair??
0
y its nt anti symmetric ??
0
Because it contains 2,3 as well as 3,2
0
if aRb and bRa ... then a=b .. this is anti symmetric right ??
0
Yes true.
0
then (2,3) and (3,2) is in relation... so cant we say that its an anti symmetric relation ??
+1
anti symetric allows only self loop and nothing else.
0
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.
0
so one can conclude that anti symmetric is one kind reflexive relation right ??
0
reflecixe relations have all the reflexive pair
but antisymetric doesnot need that property.
0
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.
0
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
0
It is not transitive as (1,1) is present and (2,2) is present but (1,2) is not present
0
@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

answered by Boss (23.9k points)
–3 votes

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

answered by Loyal (6.8k points)


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