Answer
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 the union contains a 1 on the $i$th bit as well.
If both string contains a 0 on the $i$th bit, then the union contains a 0 on the $i$th bit as well.
Intersection: $A\cap B$ =All elements that are in both A AND B
If both the string contains a 1 on the $i$th bit, then the intersection contains a 1 on the $i$th bit as well.
If either string contains a 0 on the $i$th bit, then the intersection contains a 0 on the $i$th bit as well.
Work Step by Step
Let there be n sets.
We require that the universal set is finite.
If the universal set U contains m elements, then the bit string corresponding with every set will contain m elements.
If the $i$th element of the universal set U is in the set, then the $i$th bit of the string is a 1.
If the $i$th element of the universal set U is not in the set, then the $i$th bit of the string is a 0.
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 the union contains a 1 on the $i$th bit as well.
If both string contains a 0 on the $i$th bit, then the union contains a 0 on the $i$th bit as well.
Intersection: $A\cap B$ =All elements that are in both A AND B
If both the string contains a 1 on the $i$th bit, then the intersection contains a 1 on the $i$th bit as well.
If either string contains a 0 on the $i$th bit, then the intersection contains a 0 on the $i$th bit as well.
