Set Theory Help I have a test coming up on set theory, so the basic definitions, set operations, relations and functions. 
On the test I will have to do proofs involving all of these (subset, unions, intersections, relative complements, cartesian products, different kinds of relations, partial order/strict order, inverse, composition, fuctions, one-to-one functions, onto funtions, etc.)
What I was wondering is if anyone has any tips for doing proofs. Usually I have a problem getting started with the proof...what rules to start with and what exactly I want to end up with.
Then I think of using certain theorems but when I look at solutions from my teacher he uses different theorems. 
Any tips would be great
 A: You should start by fully understanding what you have to prove. Then you have to fully understand the definitions and theorems related to the proposition. Then you should begin by unwinding the definitions using the above tools, and then derive the wanted conclusion. 
This is the only general approach I can recommend. Mathematics is about creativity, not about algorithmic solutions. 
A: The key to writing proofs is to "let the definitions do the work", but to do so, you need to have a firm grasp of the definitions you've learned, and how/when to unpack and apply them: e.g., what does it mean to say one set is a subset of another subset? Similarly for the definitions of the union and intersection of a set, the relative complement of a set (set minus), etc. 
In addition to having a solid grasp of the definitions you need to know, keep in mind that any "givens" you have to work with are also tips that you'll probably need to use in the ensuing proof.
One tip: You'll want to keep in mind that typically one proves that two sets are equal by showing each is the subset of the other, i.e., $A = B \iff A\subseteq B \;\text{ and}\;\;B\subseteq A.$ 
Another tip: To show that the cardinality of one set is equal to that of another, you can show that there exists a bijective (one-to-one & onto) function that maps the elements of one set to the other. Conversely, if you know that there exists a one-to-one correspondence between two sets, you know that the sets are equivalent, in terms of cardinality. Do you see how, in this case, you'd need the definitions of what it means to be a "function" that is "one-to-one" and "onto".
There's really no shortcut to having a firm grasp of the definitions you've learned.
Simply "memorizing" definitions can easily lead to confusion (going blank); and in any case, such rote "learning" is quickly forgotten. Try to understand the definitions, using examples, when necessary, until they make sense, and more, try to understand how those definitions relate with one another, and with what you've already learned.

There are many ways to approach a proof, in terms of "direction." Direct proof, proof by contradiction, inductive proofs: you'll want to develop facility and flexibility with such strategies.  Note that there is no "right" solution/approach - in terms of proving a statement - as there is with arithmetic calculations, or finding a derivative. 
A: Try to recollect the relevant laws and then eliminate the ones which do not have direct application  thereafter use the concept of equality of two sets or use the unknown element method  remember maths is about pattern
