Let R be a relation on a collection of sets defined as follows,

R={(A,B)|A⊆B}

Then pick out the correct statement(s).

1) R is reflexive and transitive

2) R is symmetric

3) R is antisymmetric.

4) R is reflexive but not transitive

Question

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Respuesta :

berno berno · Answer 1 · 2020-10-16T11:01:56+02:00

Answer:

Options 1) and 3) are correct.

Step-by-step explanation:

R={(A,B)|A⊆B}

Reflexive:

As A⊆A, [tex](A,A)[/tex]∈ R.

So, R is reflexive

Symmetric:

Let [tex](A,B)[/tex]∈ R. So, A⊆B

Take [tex]A=\{1,2\}\,,\,B=\{1,2,3,4\}[/tex]

Here, A⊆B but B⊄A

So, [tex](B,A)[/tex]∉ R

R is not symmetric

Transitive:

Let [tex](A,B)[/tex]∈ R and [tex](B,C)[/tex]∈ R

So, A⊆B and B⊆C.

Therefore, A⊆C

So,

[tex](A,C)[/tex]∈ R

Hence, R is transitive.

Option 1) is correct.

Antisymmetric:

Let (A,B)∈R and (B,A)∈R

So, A⊆B and B⊆A

Hence, A = B

So, R is antisymmetric

Option 3) is also correct.