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2 April, 19:49

In each case below, a relation on the set {1, 2, 3} is given. Of the three properties, reflexivity, symmetry, and transitivity, determine which ones the relation has. Give reasons.

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  1. 2 April, 20:10
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    a. is symmetric but not reflexive and transitive

    b. is reflexive and transitive but not symmetric

    c. is reflexive, symmetric and transitive

    Step-by-step explanation:

    The cases are missing in the question.

    Let the cases be as follows:

    a. R = { (1, 3), (3, 1), (2, 2) }

    b. R = { (1, 1), (2, 2), (3, 3), (1, 2) }

    c. R = ∅

    R is defined on the set {1, 2, 3}

    R is reflexive if for all x in {1, 2, 3} xRx R is symmetric if for all x, y in {1, 2, 3} if xRy then yRx R is transitive if for all x, y, z in {1, 2, 3} if xRy and yRz then xRz

    a. R = { (1, 3), (3, 1), (2, 2) } is

    not reflexive since for x=1, (1,1) is not in R symmetric since for all x, y in {1, 2, 3} if xRy then yRx not transitive because (1, 3), (3, 1) is in R but (1,1) is not.

    b. R = { (1, 1), (2, 2), (3, 3), (1, 2) } is

    is reflexive because (1, 1), (2, 2), (3, 3) is in R

    is not symmetric because for (1,2) (2,1) is not in R

    is transitive becaue for (1,1) and (1,2) we have (1,2) in R

    c. R = ∅ is

    reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.
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