Given the following definitions of properties: Reflexive means (x) (xS (x,x)r) s

Question

Given the following definitions of properties:

Reflexive means (x) (xS (x,x)r)
symmetric means (x)(y) (xS yS (x,y)r (y,x)r)
antisymmetric means (x)(y) (xS yS (x,y)r (y,x)r x = y)
transitive means (x)(y)(z) (xS yS zS (x,y)r (y,z)r (x,z)r)

where r is our relation

Let S = {0,1,2,4,6}.
Test the following binary relations on S for reflexivity, symmetry, antisymmetry, and transitivity.
(write reflexive, symmetric, antisymmetric, or transitive if the relation has that property in that order. Leave the property out if it does not have it. Use comma separators and no spaces)

a. = {(0,0), (1,1), (2,2), (4,4), (6,6), (0,1), (1,2), (2,4), (4,6)}
b. = {(0,1), (1,0), (2,4), (4,2), (4,6), (6,4)}
c. = {(0,1), (1,2), (0,2), (2,0), (2,1), (1,0), (0,0), (1,1), (2,2)}
d. = {(0,0), (1,1), (2,2), (4,4), (6,6), (4,6), (6,4)}
e. = {}

Explanation / Answer

given the following definaion

symmetric means:-

a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a.

antisymmtric means:

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anti-symmetric precisely if for all a and b in X

if R(a,b) and R(b,a), then a = b,

or, equivalently,

if R(a,b) with a b, then R(b,a) must not hold

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your answer:-

A) reflexive,antisymmtric

B)symmetry

c)symmtric,transitive

D)reflixive,symmtry

E)symmtry,antisymmtry,transitive,

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