**ind the smallest relation containing the relation {(1, 2), (1, 4), (3 , 3), (4, 1)} that is a) reflexive and transitive. b) symmetric and transitive. c) reflexive, symmetric, and transitive.**

The **Answer to the Question**

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**Here's the Solution to this Question**

*Reflexive*: For every element a in set, (a, a) must be in relation

*Symmetric*: If (a, b) is in relation, then should be (b, a)

*Transitive*: If (a, b) and (b, c) is in the relation then should be (a, c)

a. Reflexive and transitive

Given = {(1,2), (1,4), (3,4), (4,1)}

To be reflexive, it should have {(1,1), (2,2), (3,3), (4,4)}

To be transitive, it should have {(4,2), (3,2)}

Answer = {(1,1), (2,2), (3,3), (4,4), (1,2), (1,4), (3, 1), (3,2), (3,4), (4,1), (4,2)}

b. Symmetric and transitive

Given = {(1,2), (1,4), (3,4), (4,1)}

Answer = {(1,2), (1,3), (1,4), (2,1), (2, 3), (2, 4), (3,1), (3, 2), (3,4), (4,1), (4,2) (4,3)}

c.

Given {(1,2), (1,4), (3,4), (4,1)}

Answer = {(1,1), (2,2), (3,3), (4,4), (1,2), (1,3), (1,4), (2,1), (2, 3), (2, 4), (3,1), (3, 2), (3,4), (4,1), (4,2) (4,3)}