Answer true or false: For A and B sets, A ∩ B = B ∩ B'

Question

Answer true or false: For sets $A$ and $B:$ $A \cap B = B \cap B'.$

The statement is false. Let $A$ and $B$ be non-empty sets with $A = B$ and let $X = \{ a , b , c \}.$ Then

$A \cap B = \{ a \} \cap \{ a \} = \{ a \} $ and $B \cap B'= \{ a \} \cap \{ b , c\}.$ Since for all set $A, \emptyset \subseteq A$, note that $\{ a \} \cap \{ b , c \} = \emptyset.$

But then $A \cap B \neq B \cap B'$ because $\{a\} \neq \emptyset.$

Is my answer correct? This is an exercise taken from my workbook.

Answer 1

You're right but it needs to be written in a slightly better manner. For instance, you never write what $A$ actually is.

Let $X=\{a,b,c\}$ and $A=B= \{a\}.$ Then $$A\cap B=\{a\}\cap \{a\}=\{a\}$$ and $$B \cap B^c =\emptyset$$ and so $$A\cap B \neq B\cap B^c.$$

Answer 2

Yup, this is correct. Though one shouldn't need to make an example to see it's false already: $B \cap B' = \emptyset$ regardless of what $B$ is, so the only way it could be true is if $A \cap B = \emptyset$ always, but that's obviously not always the case.

Though a minor typo, I believe you meant to say $A \cap B = \{a\} \cap \{a\}$ at the start. Same result though.