1.2k views

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

edited | 1.2k views

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.
by Veteran (60k points)
edited
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I am not getting 4,4 not present means not reflexive but 1 1 is there so not irreflexive ...plz explain.
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a reflexive relation must contain all elements of the form (x,x) and an irreflexive relation should not contain any such pair.
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for element (2,4) transistive pair??
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y its nt anti symmetric ??
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Because it contains 2,3 as well as 3,2
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if aRb and bRa ... then a=b .. this is anti symmetric right ??
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Yes true.
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then (2,3) and (3,2) is in relation... so cant we say that its an anti symmetric relation ??
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anti symetric allows only self loop and nothing else.
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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.
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so one can conclude that anti symmetric is one kind reflexive relation right ??
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reflecixe relations have all the reflexive pair
but antisymetric doesnot need that property.
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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.
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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
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It is not transitive as (1,1) is present and (2,2) is present but (1,2) is not present
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@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).

option b

by Boss (34.6k points)

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

by Loyal (7.6k points)

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