Solution to For each relation below, determine if they are reflexive, symmetric, anti-symmetric, and transitive. (a) X= … - Sikademy
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Archangel Macsika

For each relation below, determine if they are reflexive, symmetric, anti-symmetric, and transitive. (a) X= { 1, 2, 3, 4} R1={(1, 2),(2, 3),(3, 4)} (b) X = {a, b, c, d, e} R1 = { (a, a), (a, b), (a, e), (b, b), (b, e), (c, c), (c, d), (d, d), (e, e) } (c) X= { 1, 2, 3, 4} R1= {(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(3, 4)}

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A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.


A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A.


A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is

called antisymmetric.


A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.



(a)

X= \{ 1, 2, 3, 4\}

R1=\{(1, 2),(2, 3),(3, 4)\}

Not reflexive because we do not have (1, 1),(2,2), (3, 3), and (4, 4).

Not symmetric because while we have (1, 2), we do not have (2, 1).

Antisymmetric.

Not transitive because we do not have (1, 3) for (1, 2) and (2, 3).  


(b)

X= \{ a, b, c, d, e\}

R1=\{(a, a), (a, b), (a, e), (b, b), (b, e), (c, c), (c, d),

(d, d), (e, e) \}

Reflexive because we have (a, a), (b, b),(c, c), (d, d), and (e,e).

Not symmetric because while we have (a, b), we do not have (b, a).

Antisymmetric.

Transitive.  


(c)

X= \{ 1, 2, 3, 4\}

R1=\{(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(3, 4)\}

Not reflexive because we do not have (1, 1),(2,2), (3, 3), and (4, 4).

Not symmetric because while we have (1, 4), we do not have (4, 1).

Not antisymmetric because we have both (1,3) and (3, 1).

Not transitive because we do not have (1, 1) for (1, 3) and (3, 1).  


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Question ID: mtid-5-stid-8-sqid-1221-qpid-959