# Equivalence relationships and binary relations

• May 23rd 2012, 04:56 PM
simmy
Equivalence relationships and binary relations
Let p be a binary relation defined on the integers Z by:

apb if and only if a2 =logically equivalent to= b2(mod7)

Prove carefully that p is an equivalence relation on Z and give the equiv classes.
__________________________________________________ ____________________
Let X = {p, q}
write down the set X x X
list all the binary relations on X which are reflexive
A binary relation p on a set X is said to be irrelfexive if (x, x) not-element p for all x element X
Write down all binary relations on the set X above which are irreflexive, and also irreflexive and symmetric simultaneously

Ok, So i started with the first part and proved it by
Took each number and squared it. The ones with the same square mod 7 are in the same equivalence class.

0^1 ≡ 0(mod7)
1^2 ≡ 1(mod7), 6
2^2 ≡4(mod7), 5
3^2 ≡2(mod7), 4

Therefore equivalence classes are {0}, {1,6} , {2,5} and {3,5} I think?(clarification would do wonders here)

However I am stuck from the X = {p, q} bit... and everything that follows any help would be much appreciated...

Thanks Simmy
• May 23rd 2012, 06:20 PM
emakarov
Re: Equivalence relationships and binary relations
Quote:

Originally Posted by simmy
apb if and only if a2 =logically equivalent to= b2(mod7)

Logical equivalence refers to propositions, not numbers. Here a2 and b2 are said to be congruent modulo 7.

Quote:

Originally Posted by simmy
Therefore equivalence classes are {0}, {1,6} , {2,5} and {3,5}

The last class should be {3, 4}. In fact, since the relation is on \$\displaystyle \mathbb{Z}\$ and not \$\displaystyle \mathbb{Z}_7\$, equivalence classes should be infinite. Also, you have not shown that p is an equivalence relation.

Quote:

Originally Posted by simmy
Let X = {p, q}
write down the set X x X

X x X = {(p, p), (p, q), (q, p), (q, q)}.

Quote:

Originally Posted by simmy
list all the binary relations on X which are reflexive

A relation on X is a subset of X x X. A reflexive relation must contain (p, p) and (q, q), but it may or may not contain other pairs.
• May 23rd 2012, 08:43 PM
simmy
Re: Equivalence relationships and binary relations