Question on Sets

[gardenia]

Let U= {q,r,s,t,u,v,w,x,y,z}
A = {q,s,u,w,y}
B= {q,s,y,z}
C= {v,w,x,y,z}

(A union B')' intersect C'

Would this end up being a null set. I'm really confused on this problem with the 3 primes.

 

[Deleted member 4993]

Let U= {q,r,s,t,u,v,w,x,y,z}
A = {q,s,u,w,y}
B= {q,s,y,z}
C= {v,w,x,y,z}

(A union B')' intersect C'

Would this end up being a null set. I'm really confused on this problem with the 3 primes.

What are the elements of set B' ....? → B' = {r,t,u,v,w,x}

What are the elements of set C' ....?

Then find A union B'

and so on.....

 

[pka]

(A union B')' intersect C'

\(\displaystyle (A\cup B')'=A'\cap B\), so \(\displaystyle (A \cup B')' \cap C'=A'\cap B\cap C'\)

 

[HallsofIvy]

\(\displaystyle (A\cup B')'=A'\cap B\), so \(\displaystyle (A \cup B')' \cap C'=A'\cap B\cap C'\)

True, and the easiest way to do this problem. But I suspect it would confuse gardenia even more. Probably better, as Subhotosh Khan suggested, to do each part, step by step.

 

[soroban]

Hello, gardenia!

\(\displaystyle U\:=\: \{q,r,s,t,u,v,w,x,y,z\}\)
\(\displaystyle A\:=\:\{q,s,u,w,y\}\)
\(\displaystyle B\:=\: \{q,s,y,z\}\)
\(\displaystyle C\:=\:\{v,w,x,y,z\}\)

\(\displaystyle \text{Find: }\: (A \cup B')' \cap C'\)

Would this end up being a null set? . Yes!

Let's do this one step at a time . . .


\(\displaystyle A \:=\:\{q,s,u,w,y\}\)
\(\displaystyle B' \:=\: \{q,s,y,z\}' \:=\:\{r,t,u,v,w,x\}\)

\(\displaystyle \text{Then: }\:A \cup B' \:=\:\{q,s,u,w,y\} \cup \{r,t,u,v,w,x\} \:=\:\{q,r,s,t,u,v,w,x,y\}\)

\(\displaystyle \text{Hence: }\: (A\cup B')' \:=\:\{q,r,s,t,u,v,w,x,y\}' \:=\:\{z\}\)


\(\displaystyle C' \:=\:\{v,w,x,y,z\}' \:=\:\{q,r,s,t,u\}\)


\(\displaystyle \text{Therefore: }\: (A \cup B')' \cap C' \:=\: \{z\} \cap \{q,r,s,t,u\} \:=\:\emptyset \)