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).
R is reflexive and transitive
R is symmetric
R is antisymmetric
R is reflexive but not transitive.

Answer 1

SOLUTION :

GIVEN

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

$\sf{R = \{ (A, B) \: : A \subseteq \: B \}}$

TO CHOOSE THE CORRECT OPTIONS

• R is reflexive and transitive

• R is symmetric

• R is antisymmetric

• R is reflexive but not transitive

EVALUATION

Let U be the Universal set, the collection of all sets for the given problem

CHECKING FOR REFLEXIVE

$\sf{ Let \: \: A \in \: U\: }$

$\sf{Then \: \: A \subseteq \: A}$

$\sf{So \: \: (A, A) \in R}$

So R is reflexive

CHECKING FOR SYMMETRIC

Let A = { 1,2 } and B = { 1,2, 3,4 }

Then A is a subset of B but B is not a subset of A

So R is not symmetric

CHECKING FOR ANTISYMMETRIC

$\sf{ Let \: \: A, B \in U\: }$

$\sf{Such \: that \: (A, B) \in R \: \: and \: \: (B, A) \in R}$

Which gives

$A \subseteq \: B \: \: and \: \: B \subseteq \: A$

Which gives A = B

SO R is antisymmetric

CHECKING FOR TRANSITIVE

$\sf{Let \: \: A, B, C \in \: U}$

$\sf{Such \: that \: (A, B ) \in \: R \: and \: (B, C) \in R}$

$\sf{Then \: \: A \subseteq \: B \: \: and \: \: B \subseteq \: C \: }$

Which together gives

$\sf{A \subseteq \: C}$

$\implies \: \sf{ (A, C ) \in \: R }$

So R is transitive

FINAL RESULT

• R is reflexive and transitive

• R is antisymmetric

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