Okay, so this is more of a question of how to tackle certain problems. I need a

Question

Okay, so this is more of a question of how to tackle certain problems. I need a very thorough guide on how to prove ALL types of relation, one to one, onto. And what i mean by that is, I need to know how to do

Relation types of problems:

1. Is it a relation equivalence?

2. Prove it is reflexive, symmetric, transitive.

3. Disprove it is reflexive, symmetric, transitive.

Functions:

1. is it one to one? is it onto?

2. Prove it is one to one. Prove it is onto.

3. Disprove it is one to one. disprove it is onto.

Sets:

example (because i don't know how to explain it)

Let E be the set of all even integers, O be the set of all odd integers. Let X = {n in Z(0-infinity) | n = x + y for some x, y in O}

a. prove X is in E

b. prove E is in(replacement for the symbol) X

I need to know how to do ANY problem that i might encounter with this ^ format.

Set builder notation:

I need a decent explanation on how to tackle problems where they say "write statement in set builder notation".

Proofs:

Any other kind of proofs that you can think of that will be in an introductory Discrete math course. You do not have to solve this problem, but this is the hardest kind of proof that our teacher will be giving us:

Extra problem: Let G be a simple graph with n vertices. Prove that there must be two vertices with the same degree. (Equivalently, prove that degree of vertices cannot be all distinct)

Give me a good explanation on how to tackle ^.

I know it's a lot, but if it's a really good explanation I'll even create a new question that won't mean anything, send you the link, and you just post in it and i'll give you an extra 1500 points just for getting this done.

And please get it done ASAP i have a midterm in 3 hours.

Explanation / Answer

A given relation on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.

Reflexive:-

A reflexive relation is a binary relation on a set for which every element is related to itself. In other words, a relation on a set S is reflexive when x ~ x holds true for every x in S.

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself.

Symmetric:-

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

Transitive:-

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.

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