So our book for class only talks about sets-within-sets for only a brief moment, and only gives the example of the empty set being an element of a set of the empty set (If A = {$\phi$} and B = {{$\phi$}} then A $\in$ B but A $\nsubseteq$ B).
But let's say that A = { 3, {4, 5}, 7, 8 } and B = {4, {5, 7}, 6, 8}.
What would A $\cap$ B be?
A $\cup$ B?
A – B?
My personal guesses –
Intersection: {8} ;
Union: {3, {4,5}, {5,7}, 6, 7, 8} ;
A – B: {3, {4,5}, 7}
Best Answer
Recall that $$A \cap B = \{x : x \in A \;\text{ and }\; x\in B \} \\ A \cup B = \{x : x \in A \;\text{ or }\; x\in B \} \\ A - B = \{x: x \in A \; \text{and} \; x\not\in B \}. $$
In particular, if $A = \{ 3, \{4, 5\}, 7, 8 \}$ and $B = \{4, \{5, 7\}, 6, 8\}$, then we have that $$A \cap B = \{8\} \\ A \cup B = \{3,4,6,7,8,\{4,5\},\{5,7\}\} \\ A - B = \{3,\{4,5\},7\}. $$