Question

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.

please give right answer. best and fast would be marked brainliest!!

Answer 1

4) R Is Reflexive But Not Transitive.

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Answer 2

The correct statement is: R is reflexive and transitive.

The explanation is as follows:

Reflexive: A relation R on a set S is reflexive if for every element x in S, (x,x) is in R. In this case, for any set A, A ⊆ A is always true, so R is reflexive.

Symmetric: A relation R on a set S is symmetric if for every pair of elements (x,y) in S, (y,x) is also in R. In this case, if A ⊆ B, it does not necessarily mean that B ⊆ A, so R is not symmetric.

Antisymmetric: A relation R on a set S is antisymmetric if for every distinct pair of elements (x,y) in S, if (x,y) is in R then (y,x) is not in R. In this case, if A ⊆ B and B ⊆ A, then A = B, so R is antisymmetric.

Transitive: A relation R on a set S is transitive if for every three elements x, y, and z in S if (x,y) is in R and (y,z) is in R, then (x,z) is in R. In this case, if A ⊆ B and B ⊆ C, then A ⊆ C, so R is transitive.

Therefore, R is reflexive and transitive, but not symmetric or antisymmetric.

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