This discussion focuses on proving set theory statements, specifically: (a) if \( A \cap B = \emptyset \), then \( (A \times C) \cap (B \times C) = \emptyset \), and (b) \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \). Participants share techniques for formal proofs, emphasizing the importance of assuming contrary conditions and using logical deductions. The conversation highlights the need for clarity in reasoning and the structure of proofs, particularly in set theory.
PREREQUISITESMathematics students, educators, and anyone interested in mastering formal proof techniques in set theory and enhancing their logical reasoning skills.
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)