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)
Proving sets is a fundamental aspect of mathematics that allows us to establish the truth or validity of mathematical statements. It helps us to logically and rigorously demonstrate that a particular statement or relationship between sets is always true, regardless of specific examples or cases.
There are several techniques that can be used in proving sets, including direct proof, proof by contradiction, proof by induction, and proof by contrapositive. Each technique has its own strengths and is chosen based on the nature of the statement being proved.
Sure, a direct proof involves starting with the given assumptions and using logical steps to arrive at the desired conclusion. For example, to prove that the sum of two even numbers is always even, we can start by assuming that a and b are even numbers. Then, we can express them as a = 2m and b = 2n, where m and n are integers. Thus, the sum a + b = 2m + 2n = 2(m + n), which is also an even number.
Proof by contradiction is a technique where we assume the opposite of what we want to prove and then show that it leads to a contradiction. This proves that the original statement must be true. This technique is often used when direct proof is not possible or when the statement is negative in nature.
Proof by induction is a technique commonly used for proving statements that involve a variable, such as n, where n is a positive integer. The proof involves showing that the statement is true for the base case, usually n = 1, and then assuming it is true for some arbitrary value of n. By using this assumption, we can show that the statement is also true for n+1. This proves that the statement is true for all positive integers n.