Question 143096
Start with the given sets:


*[Tex \LARGE A=\left\{b,c,d\right\}]


*[Tex \LARGE B=\left\{c,e,f,g\right\}]



Find their respective complement sets:


*[Tex \LARGE A^\prime=\left\{a,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}]


*[Tex \LARGE B^\prime=\left\{a,b,d,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}]




a)


So 


*[Tex \LARGE A \cup B^\prime=\left\{b,c,d,a,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}]


Remember, a union is simply the combination of the unique elements from both sets




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b)


From part a), we got *[Tex \LARGE A \cup B^\prime=\left\{b,c,d,a,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}]




So 


*[Tex \LARGE \left(A \cup B^\prime\right) \cap B=\left\{b,c,d,a,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}\cap \left\{c,e,f,g\right\}=\left\{c\right\}]


Remember, to find the intersection of two sets, simply look for any elements that the two sets have in common




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c)


First find  *[Tex \LARGE A \cup B]



*[Tex \LARGE A \cup B=\left\{b,c,d,e,f,g\right\}]




So 



*[Tex \LARGE \left(A \cup B\right)\cap\left(A \cup B^\prime\right)=\left\{b,c,d,e,f,g\right\}\cap\left\{b,c,d,a,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\right\}=\left\{b,c,d\right\}]