Answer
a)$A\cup B$={$a,b,c,d,e,g,p,t,v$}
b)$A\cap B$={$b,c,d$}
c)$(A\cup D)\cap (B \cup C)$={$b,c,d,e,i,o,t,u,x,y$}
d)$A\cup B\cup C\cup D$={$a,b,c,d,e,g,h,i,n,o,p,t,u,v,x,y,z$}
Work Step by Step
Given :
A = {a, b, c, d, e},
B = {b, c, d, g, p, t, v},
C = {c, e, i, o, u, x, y, z}, and
D = {d, e, h, i, n, o, t, u, x, y}.
If the $i$th bit in the string is 1, then the $i$th letter of the alphabet is in the set.
If the $i$th bit in the string is 0, then the $i$th letter of the alphabet is NOT in the set.
The alphabet contains 26 letters, thus each string needs to contain 26 bits.
A: 11111 00000 00000 00000 00000 0
B: 01110 01000 00000 10001 01000 0
C: 00101 00010 00001 00000 10011 1
D: 00011 00110 00011 00001 10011 0
a) Union: $A \cup B$: All elements that are either in A or in B
If either string contains a 1 on the $i$th bit, then $A \cup B$ contains a 1 on the $i$th bit as well.
$A\cup B$:11111 01000 00000 10001 01000 0
This string then corresponds to the set:
$A\cup B$={$a,b,c,d,e,g,p,t,v$}
b) Intersection:$A \cap B$: All elements that are both in A AND B
If both strings contains a 1 on the $i$th bit, then $A \cap B$ contains a 1 on the $i$th bit as well.
$A\cap B$:01110 00000 00000 00000 00000 0
This string then corresponds to the set:
$A\cap B$={$b,c,d$}
c) If either string contains a 1 on the $i$th bit, then union contains a 1 on the $i$th bit as well.
$A\cup D$:11111 00110 00011 00001 10011 0
$B\cup C$: 01111 01010 00001 10001 11011 1
If both strings contains a 1 on the $i$th bit, then intersection contains a 1 on the $i$th bit as well.
$(A\cup D)\cap (B \cup C)$:01111 00010 00001 00001 10011 0
This string then corresponds to the set:
$(A\cup D)\cap (B \cup C)$={$b,c,d,e,i,o,t,u,x,y$}
d)If either string contains a 1 on the $i$th bit, then union contains a 1 on the $i$th bit as well.
$A\cup D$:11111 00110 00011 00001 10011 0
$B\cup C$: 01111 01010 00001 10001 11011 1
Since the union of commutative:$A\cup B\cup C\cup D=(A\cup D)\cup (B \cup C)$
$A\cup B\cup C\cup D$:11111 01110 00011 10001 11011 1
This string then corresponds to the set:
$A\cup B\cup C\cup D$={$a,b,c,d,e,g,h,i,n,o,p,t,u,v,x,y,z$}
