Given the relation $\{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}$ determine whether it is reflexive, transitive, symmetric, or anti-symmetric.

Question

Given the relation $\{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}\,$ determine whether it is reflexive, transitive, symmetric, or anti-symmetric.

I found this set to be reflexive and symmetric. But not transitive and anti-symmetric.

Would it be correct to say that this set would be anti-symmetric if we remove either the element $(1,2)$ or $(2,1)$?

Also, the solution claims this set to be transitive. But I found it not to be so, due to the reasoning that $(2,3)$ and $(3,4)$ is not in the set.

Is my understanding of these ideas correct? Thank you.

Answer 1

It is reflexive if this is a relation over the set $\{1,2,3,4\}$, and yes, the relation is symmetric.

Yes, If we remove $(1,2)$ or $(2,1)$ then it is anti-symmetric.

The relation is transitive, we do not need $(2,3)$ and $(3,4)$ to be in the set. Especially there is no pairs in the relation $(2,x)$ and $(x,3)$, which is what we would need in order to force $(2,3)$ to be in the relation due to transitivity.

Answer 2

is reflexive as (a,a);a=1(1)4 exist,is symmetric as (1,2) nd (2,1) exist and transitive as corrosponding term of (2,1),(1,2)- (1,1) exist in this relation.

Answer 3

It is reflexive as {(1,1),(2,2),(3,3),(4,4)} are present in the set assuming it is a relation of {1,2,3,4} (original set).

Taking (1,1) as an example.

All (x,x) are symmetric too as: if x = 1, y = 1. (x,y) ∈ R and (y,x) ∈ R.

All (x,x) are symmetric too as: if x = 1, y = 1, z = 1 (x,y) ∈ R, (y,z) ∈ R and (x,z) ∈ R.

Here, (1,2) and (2,1) ∈ R, so the only other value that is not reflexive is taken care of by being symmetric.

Taking, (1,2)∈ R, (2,1) ∈ R, (1,1) ∈ R.
taking (2,1) ∈ R, (1,2) ∈, (2,2) ∈ R.

Checking for you, you will see all values have transitive relations existential.