Question 1196537
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There's ambiguity when you mention "with a line on top"


Is the line solely over the B? Or is it over everything?


Do you mean *[tex \Large A \cap \overline{B}]


Or do you mean *[tex \Large  \overline{A \cap B}]


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I'll go over each interpretation.


Let's focus on *[tex \Large A \cap \overline{B}]


The universal set is 
U = {q, r, s, t, u, v, w, x, y, z}
Cross off the stuff you find in set B
So we'll cross off: q, s, y, and z
This forms the set B' which is the same as writing *[tex \Large \overline{B}]
B' = {r, t, u, v, w, x}
This is the complement of set B.


Stuff in B' is not found in B, and vice versa.
They are complete opposite of one another.
The two sets union together to form the universal set.


Think of it like this:
stuff inside B is inside a house
stuff outside B is outside the house


Now we'll intersect the set A with set B'
A = {q, s, u, w, y}
B' = {r, t, u, v, w, x}
We see that the letters u and w are inside both A and B' at the same time. 


Hence A ∩ B' = {u, w}
which is the same as saying *[tex \Large A \cap \overline{B} = \{ u, w \}]


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Now onto *[tex \Large  \overline{A \cap B}]


This is the same as writing (A ∩ B)'
We're taking the complement of set (A ∩ B)


Here are sets A and B
A = {q, s, u, w, y}
B = {q, s, y, z}
which means,
A ∩ B = {q, s, y}
since those three letters are found in both sets


We'll delete q,s,y from the universal set
U = {q, r, s, t, u, v, w, x, y, z}
(A ∩ B)' = {<font color=red><s>q</s></font>, r, <font color=red><s>s</s></font>, t, u, v, w, x, <font color=red><s>y</s></font>, z}
(A ∩ B)' = {r, t, u, v, w, x, z}


Therefore, *[tex \Large  \overline{A \cap B} = \{r, t, u, v, w, x, z\}]
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