# Proof verification of union/intersection of family of sets

by user707991   Last Updated October 17, 2019 18:20 PM

Can I please get a proof verification or a showing of how to prove these set equalities correctly?

Let $$B$$ be an arbitrary set.$$C_{b}=B-\{b\}$$

Prove:

(1)If $$B=\emptyset$$ or $$|B|=1$$, $$\bigcup\limits_{b\in B}C_{b}= \emptyset$$,

(2)Prove if $$|B|>1$$,$$\bigcup\limits_{b\in B}C_{b}=B$$,

(3)Prove $$\forall B$$, $$\bigcap\limits_{b\in B}C_{b}=\emptyset$$

Proof(1):

If $$B=\emptyset$$

Then $$\bigcup\limits_{b\in B}\emptyset= \emptyset$$

If $$|B|=1$$, $$\bigcup\limits_{b\in B}C_{b}= \emptyset$$

Assume By way of contradiction(BWOC) $$y \in C_b$$ for some $$b\in B$$

then $$y \in B-\{y\}$$ and $$B=\{y\}$$ Since $$|B|=1$$

this contradicts the assumption that $$y \in C_b$$ for some $$b\in B$$.

Proof(2):

Assume $$|B|>1$$ then $$\bigcup\limits_{b\in B}C_{b}= B$$

Since $$C_b \subset B$$ for all $$b$$ this is proved.

Assume $$y\in B$$

let $$b\neq y$$ then $$y \in C_b$$ for $$b\neq y$$

thus $$y\in \bigcup\limits_{b\in B}C_{b}$$

Proof(3):

Prove $$\bigcap\limits_{b\in B}C_{b}=\emptyset$$

Assume BWOC $$y \in \bigcap\limits_{b\in B}C_{b}$$

since $$y \in C_b$$ for all $$b$$, and $$y\in B$$,

$$y \in B-\{y\}$$ this contradicts the assumption so $$\bigcap\limits_{b\in B}C_{b}= \emptyset$$

Tags :

## Related Questions

Updated November 18, 2017 02:20 AM

Updated August 18, 2016 08:08 AM

Updated April 15, 2017 20:20 PM

Updated April 18, 2017 04:20 AM