Given that the universal set is $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, we are asked to represent the sets $A$, $B$, and their complements and intersections in terms of bit strings. A bit string uses 0 or 1 to indicate the absence or presence of elements corresponding to their positions in the universal set.

Let’s break it down and find the bit strings for each set and operation:

Step 1: Bit Strings for A, B, and Complements

• Universal set $U$ has 8 elements, so the bit strings will be 8 bits long.

• Set $A = \{1, 2, 4, 6\}$:

  • $A$ includes elements 1, 2, 4, and 6. In terms of a bit string, this is:
    $A = 11010100$
    (1s at positions corresponding to 1, 2, 4, and 6, and 0s elsewhere).

• Set $B = \{2, 3, 5, 7\}$:

  • $B$ includes elements 2, 3, 5, and 7. In terms of a bit string, this is:
    $B = 01101010$
    (1s at positions corresponding to 2, 3, 5, and 7, and 0s elsewhere).

• Complement of $A$ ($A^c$):

  • $A^c$ includes all elements not in $A$, which are $\{3, 5, 7, 8\}$. The bit string for $A^c$ is:
    $A^c = 00101011$.

• Complement of $B$ ($B^c$):

  • $B^c$ includes all elements not in $B$, which are $\{1, 4, 6, 8\}$. The bit string for $B^c$ is:
    $B^c = 10010101$.

Step 2: Set Operations

• Union of $A$ and $B$ ($A \cup B$):

  • The union includes elements that are in either $A$ or $B$. The bit string for $A \cup B$ is:
    $A \cup B = 11111110$.

• Intersection of $A$ and $B$ ($A \cap B$):

  • The intersection includes elements that are in both $A$ and $B$. The bit string for $A \cap B$ is:
    $A \cap B = 01000000$.

• Union of complement of $A$ and $B$ ($A^c \cup B$):

  • This operation includes elements that are in $A^c$ or $B$. The bit string for $A^c \cup B$ is:
    $A^c \cup B = 01101011$.

• Union of complement of $B$ and $A$ ($B^c \cup A$):

  • This operation includes elements that are in $B^c$ or $A$. The bit string for $B^c \cup A$ is:
    $B^c \cup A = 11010101$.

• Union of complement of $A$ and complement of $B$ ($A^c \cup B^c$):

  • This operation includes elements that are in $A^c$ or $B^c$. The bit string for $A^c \cup B^c$ is:
    $A^c \cup B^c = 10111111$.

• Complement of intersection of $A$ and $B$ ($(A \cap B)^c$):

  • The complement of the intersection is the set of elements not in $A \cap B$. The bit string for $(A \cap B)^c$ is:
    $(A \cap B)^c = 10111111$.

Summary of Bit Strings

1. $A = 11010100$
2. $B = 01101010$
3. $A^c = 00101011$
4. $B^c = 10010101$
5. $A \cup B = 11111110$
6. $A \cap B = 01000000$
7. $A^c \cup B = 01101011$
8. $B^c \cup A = 11010101$
9. $A^c \cup B^c = 10111111$
10. $(A \cap B)^c = 10111111$

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Would you like more details on any step?
Here are some related questions:

1. How is the complement of a set represented in terms of a bit string?
2. What is the general approach for finding the union of two sets using bit strings?
3. How do you find the intersection of two sets using bit strings?
4. Can you explain how to compute the complement of a union or intersection using De Morgan’s laws?
5. How does the universal set affect the length of bit strings?

Tip: The union of sets corresponds to the logical "OR" operation, while the intersection corresponds to the logical "AND" operation on bit strings.