Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, subsets $A = \{1, 2, 4, 6\}$ and $B = \{2, 3, 5, 7\}$, we need to represent the following sets as bit strings relative to $U$:

1. Bit string representation

A bit string uses the order of the elements in $U$ to represent whether an element is in the subset (1) or not (0).

Universal Set (U): $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$ Positions: $1, 2, 3, 4, 5, 6, 7, 8$

• A: $A = \{1, 2, 4, 6\}$ $A \text{ bit string}: 11010100$

• B: $B = \{2, 3, 5, 7\}$ $B \text{ bit string}: 01101010$

2. Complement of A ($A^c$)

The complement of $A$ contains all elements of $U$ that are not in $A$.

$A^c = \{3, 5, 7, 8\}$ $A^c \text{ bit string}: 00101011$

3. Complement of B ($B^c$)

The complement of $B$ contains all elements of $U$ that are not in $B$.

$B^c = \{1, 4, 6, 8\}$ $B^c \text{ bit string}: 10010101$

4. Union of A and B ($A \cup B$)

The union of $A$ and $B$ includes all elements that are in either $A$ or $B$ (or both).

$A \cup B = \{1, 2, 3, 4, 5, 6, 7\}$ $A \cup B \text{ bit string}: 11111110$

5. Intersection of A and B ($A \cap B$)

The intersection of $A$ and $B$ includes only the elements that are in both $A$ and $B$.

$A \cap B = \{2\}$ $A \cap B \text{ bit string}: 01000000$

6. Union of the complement of A and B ($A^c \cup B$)

$A^c \cup B = \{1, 2, 3, 5, 7, 8\}$ $A^c \cup B \text{ bit string}: 11101011$

7. Union of the complement of B and A ($B^c \cup A$)

$B^c \cup A = \{1, 2, 4, 6, 8\}$ $B^c \cup A \text{ bit string}: 11010101$

8. Union of complements of A and B ($A^c \cup B^c$)

$A^c \cup B^c = \{1, 3, 4, 5, 6, 7, 8\}$ $A^c \cup B^c \text{ bit string}: 10111111$

9. Complement of the union of A and B ($(A \cup B)^c$)

The complement of the union of $A$ and $B$ includes all elements of $U$ that are not in $A \cup B$.

$(A \cup B)^c = \{8\}$ $(A \cup B)^c \text{ bit string}: 00000001$

Final Answers:

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. $A^c \cup B$: 11101011
8. $B^c \cup A$: 11010101
9. $A^c \cup B^c$: 10111111
10. $(A \cup B)^c$: 00000001

Would you like further clarification or details on any of these operations?

Here are 5 related questions for you to consider:

1. How can bit strings be used to represent larger sets efficiently?
2. What is the relationship between set operations and logical operations in computer science?
3. Can you find the intersection of the complements $A^c \cap B^c$?
4. How would the bit string representation change if $U$ had more elements?
5. Can you find the symmetric difference $A \Delta B$ and its bit string?

Tip: Understanding set operations in terms of bit strings is crucial in areas like digital logic and computer networks.