Equivalence relationships and binary relations

Let p be a binary relation defined on the integers Z by:

apb if and only if a^{2} =logically equivalent to= b^{2}(mod7)

Prove carefully that p is an equivalence relation on Z and give the equiv classes.

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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

Re: Equivalence relationships and binary relations

Quote:

Originally Posted by

**simmy** apb if and only if a^{2} =logically equivalent to= b^{2}(mod7)

Logical equivalence refers to propositions, not numbers. Here a^{2} and b^{2} 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.

Re: Equivalence relationships and binary relations

Dude your a beast, thanks