Question on testing logical truths for set operations

[schlynn]

My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this type of work they always use vein-diagrams to "prove" if two statements are the same. I personally don't like vein-diagrams, they don't feel rigourus enough I suppose, so my question is there another way to work on these problems? Like a more algebraic way I suppose, not drawing circles and checking overlapping sections, just isn't the type of math I enjoy.

 

[Stephen Tashi]

My question is on how to answer if two statements are equal in set theory.

Two "expressions" might be "equal" or "not equal" but , technically, "statements" are "equivalent" or "not equivalent".

If you want to prove that two expressions P and Q describe the same set, prove that the statement "x is a member of P " is equivalent to "x is a member of Q". For example, proving DeMorgans laws for sets can be done by using DeMorgans laws for propositions. ( such as "not (x is an elment of A or x is an element of B)" is equivalent to "x is not an element of A and x is not an element of B".