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 on a set is called reflexive if for every element
A relation on a set is called symmetric if whenever for all
A relation on a set such that for all if and then is
called antisymmetric.
A relation on a set is called transitive if whenever and then for all
(a)
Not reflexive because we do not have and
Not symmetric because while we have we do not have
Antisymmetric.
Not transitive because we do not have for and
(b)
Reflexive because we have and
Not symmetric because while we have we do not have
Antisymmetric.
Transitive.
(c)
Not reflexive because we do not have and
Not symmetric because while we have we do not have
Not antisymmetric because we have both and
Not transitive because we do not have for and