The discussion revolves around proving set-theoretic statements, specifically focusing on two claims involving intersections and unions of sets. Participants seek techniques for formal proofs and clarification on logical reasoning within set theory.
Participants generally agree on the need for formal proof techniques, but there is no consensus on the specific methods or steps to take in completing the proofs. The discussion includes various approaches and suggestions without a clear resolution.
Some participants express uncertainty about filling in logical steps in their proofs, indicating potential gaps in understanding formal proof structures. There is also a reliance on definitions of set operations that may not be fully articulated.
This discussion may be useful for students learning set theory, particularly those struggling with formal proof techniques and logical reasoning in mathematics.
a. Assume on the contrary that $(x,y)\in (A\times C)\cap (B\times C)$. Thus $x\in A\cap B$ and $y\in C$ (why?). Can you finish?paulmdrdo said:prove the followinga. prove that if $A\cap B=\emptyset$, then $(A\times C)\cap (B\times C)=\emptyset$
b. $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$
i don't have any idea how i would start proving this.
can you give me some techniques on proofs.
The 'second why' is quite straightforward. If you know that $x\in A$, then $x$ also lies in any superset of $A$. Isn't it? Thus $x\in A$ gives $x\in A\cup B$ since $A\cup B$ is a superset of $A$. Similarly $x\in A$ gives $A\cup C$ too.paulmdrdo said:caffeinemachine i know nothing about formal way of proving. i can say why that is true in words or verbally but not in a generalize way. i want a formal presentation of proofs like what you are doing. can you give some tips. because every time i encounter this kind of problems i always feel discouraged.
but this is what i tried
$x\in A\cap B$ and $y\in C$ because we assume that $(x,y)\in (A\times C)\cap (B\times C) $
i don't know howfill in the other (whys)