Are Equivalence Classes Confusing?

[shivajikobardan]

Homework Statement

Confusions about finding equivalence classes for the set of natural numbers corresponding to equivalence relation a+b is even.

Relevant Equations

N/A

 

[Mark44]

Your conclusion that $[0] \equiv [2] \equiv [4] \dots \equiv[2n] = \{2, 4, 6, 8, \dots \}$ looks OK to me, and similar for $[1]\equiv [3] \equiv [5] \dots \equiv[2n + 1] = \{1, 3, 5, 7, \dots \}$.

There are several things in what you wrote that are unclear, though.

$R = \{(a, b) : (a + b) \text{ is even} \}$

$a \in N$, but what set does b belong to?

$[a] = \{x | x \in A \land (a, x) \in R \}$

What is set A? You haven't defined it, so there's no way to determine whether an element x belongs to A or not.

 

[shivajikobardan]

Your conclusion that $[0] \equiv [2] \equiv [4] \dots \equiv[2n] = \{2, 4, 6, 8, \dots \}$ looks OK to me, and similar for $[1]\equiv [3] \equiv [5] \dots \equiv[2n + 1] = \{1, 3, 5, 7, \dots \}$.

There are several things in what you wrote that are unclear, though.

$a \in N$, but what set does b belong to?

I thought $R=\left\{(a,b):a+b~~ is ~~even\right\}$
automatically means $(a,b) \in N$ If not yes b also belongs to N.(it is not given in question though).

What is set A? You haven't defined it, so there's no way to determine whether an element x belongs to A or not.

That part is definition part. My bad I put there (thought it would be useful but turned out opposite). Here A=N in my question. It is the given set.