Sets - Difference of Sets-advanced

Let $A = \{x : x \text{ is a factor of } 12\}$ and $B = \{x : x \text{ is a factor of } 18\}$. Find $A - B$ and $B - A$.

First, list the elements of each set: $A = \{1, 2, 3, 4, 6, 12\}$ $B = \{1, 2, 3, 6, 9, 18\}$

To find $A - B$, we remove elements of $B$ from $A$: Common elements are $\{1, 2, 3, 6\}$. $A - B = \{4, 12\}$

To find $B - A$, we remove elements of $A$ from $B$: $B - A = \{9, 18\}$

The set $A - B$ contains elements that are strictly in $A$ and not in the intersection $A \cap B$. Similarly, $B - A$ contains elements strictly in $B$.

If $n(A) = 30$, $n(B) = 25$, and $n(A \cup B) = 45$, find $n(A - B)$ and $n(A \Delta B)$.

Step 1: Find $n(A \cap B)$ using the formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ $45 = 30 + 25 - n(A \cap B)$ $45 = 55 - n(A \cap B)$ $n(A \cap B) = 10$

Step 2: Find $n(A - B)$: $n(A - B) = n(A) - n(A \cap B) = 30 - 10 = 20$

Step 3: Find $n(B - A)$: $n(B - A) = n(B) - n(A \cap B) = 25 - 10 = 15$

Step 4: Find $n(A \Delta B)$: $n(A \Delta B) = n(A - B) + n(B - A) = 20 + 15 = 35$

We use the cardinality properties of sets. The symmetric difference is the sum of the cardinalities of $(A - B)$ and $(B - A)$ because they are disjoint.