Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(
a
,
b
) : both
a
and
b
are either odd or even}. Show that R is an equivalence
relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}

Question

Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(
a
,
b
) : both
a
and
b
are either odd or even}. Show that R is an equivalence
relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each
other and all the elements of the subset {2, 4, 6} are related to each other, but no
element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}

goformit100 · Answer

@terenzreignz

terenzreignz · Answer

Well this is lengthy...

terenzreignz · Answer

First one, show reflexivity... of course, a number would be related to itself, since (a,a) would mean a is either odd or even, therefore both a and a are odd or both are even.

Symmetry... If (a,b) is in the relation, then either a and b are odd or a and b are even.  therefore, (b,a) would be in the set as well

Transitivity... If (a,b) and (b,c) are in the set, then a and b have the same parity, meaning a and c have the same parity since b and c are related.  threfore, (a,c) is in the relation

thus the first relation is an equivalence relation

goformit100 · Answer

Thank you Sir

terenzreignz · Answer

Since all elements of {1,3,5,7} are odd, then they are all related, etc

terenzreignz · Answer

I'm signing off for now~
-------------------------------------------
Terence out