Let the universal set $U = \{ \text{apple, orange, pear, banana, grape, cherry, peanut} \}$. Assign increasing indices to elements of $U$ as follows:

$U = \{\text{apple (1), orange (2), pear (3), banana (4), grape (5), cherry (6), peanut (7)}\}.$

The bit string representation of a set indicates the presence ($1$) or absence ($0$) of elements of $U$ in the set.

Step 1: Representing sets as bit strings

1. $A = \{\text{apple, pear, grape, cherry}\}$:

   • Elements $1, 3, 5, 6$ are in $A$.
   • $\text{Bit string for } A = 1011010$.

2. $B = \{\text{pear, orange, cherry}\}$:

   • Elements $2, 3, 6$ are in $B$.
   • $\text{Bit string for } B = 0110010$.

Step 2: Operations on sets

(a) Union $A \cup B$:

$A \cup B = \{\text{apple, orange, pear, grape, cherry}\}.$ Bitwise OR operation: $1011010 \, \text{OR} \, 0110010 = 1111010.$

(b) Intersection $A \cap B$:

$A \cap B = \{\text{pear, cherry}\}.$ Bitwise AND operation: $1011010 \, \text{AND} \, 0110010 = 0010010.$

(c) Difference $A - B$:

$A - B = \{\text{apple, grape}\}.$ Bitwise subtraction (where $1 - 1 = 0$ and $1 - 0 = 1$): $1011010 - 0110010 = 1001000.$

(d) Symmetric Difference $A \Delta B$:

$A \Delta B = (A - B) \cup (B - A) = \{\text{apple, orange, grape}\}.$ Bitwise XOR operation: $1011010 \, \text{XOR} \, 0110010 = 1101000.$

Final Results

1. $A \cup B$: $1111010$.
2. $A \cap B$: $0010010$.
3. $A - B$: $1001000$.
4. $A \Delta B$: $1101000$.

Would you like me to clarify or expand on any specific operation?

Related Questions

1. How can these operations be expressed using Venn diagrams?
2. What is the complement of $A$ in $U$?
3. How is $(A \cup B)^C$ represented as a bit string?
4. Can we generalize this process for larger universal sets?
5. How does the result change if $U$ changes?

Tip: For bit string operations, ensure alignment of indices with the universal set