To solve the problem, we need to use the given information to find the value of $x$ and the cardinality of $A \cup B$.

First, let's draw a Venn diagram to illustrate the information.

1. Venn Diagram:

• $n(A - B) = 60 + 3x$
• $n(B - A) = 8x$
• $n(A \cap B) = x - 4$

Venn Diagram Representation

plaintextCopy code
         _________
        /         \     
       /           \    
   (60 + 3x)   (x - 4)   (8x)
     A             ∩             B

Given Information:

1. $n(A - B) = 60 + 3x$
2. $n(B - A) = 8x$
3. $n(A \cap B) = x - 4$
4. $n(A) = n(B)$

We know: $n(A) = n(A - B) + n(A \cap B)$ $n(B) = n(B - A) + n(A \cap B)$

Since $n(A) = n(B)$, we equate the expressions for $n(A)$ and $n(B)$:

$n(A - B) + n(A \cap B) = n(B - A) + n(A \cap B)$

Substitute the given values:

$(60 + 3x) + (x - 4) = (8x) + (x - 4)$

Simplify the equation:

$60 + 3x + x - 4 = 8x + x - 4$ $60 + 4x - 4 = 9x - 4$ $56 + 4x = 9x - 4$

Solve for $x$:

$56 + 4 = 9x - 4x$ $60 = 5x$ $x = 12$

a) Value of $x$: $x = 12$

b) Finding $n(A \cup B)$:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Since $n(A) = n(B)$, let's find $n(A)$ or $n(B)$:

$n(A) = n(A - B) + n(A \cap B)$ $n(A) = 60 + 3x + (x - 4)$ Substitute $x = 12$:

$n(A) = 60 + 3(12) + (12 - 4)$ $n(A) = 60 + 36 + 8$ $n(A) = 104$

Since $n(A) = n(B)$, we have: $n(B) = 104$

Now, find $n(A \cup B)$:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$ $n(A \cup B) = 104 + 104 - (12 - 4)$ $n(A \cup B) = 208 - 8$ $n(A \cup B) = 200$

Thus, the answers are: a) The value of $x$ is $12$. b) $n(A \cup B) = 200$.

Do you want more details or have any questions?

Here are some questions you might want to ask:

1. Can you explain the steps to solve for $x$ in more detail?
2. How do you derive the formula for $n(A \cup B)$?
3. What are the properties of sets used in this problem?
4. Can you provide another example similar to this problem?
5. How do the Venn diagram regions correspond to the set notation?

Tip: When solving set problems, it's helpful to draw a Venn diagram and label each region according to the given information to visualize the relationships between sets.