The set `(A cap B')cup (B cap C)` equals

Options

A. `A' cup B cup C `
B. `A' cup B `
C. `A'cup C`
D. `A' cap B`

Explanation

To solve the expression \((A \cap B') \cup (B \cap C)\), we will follow these steps:

### Step 1: Understand the expression
The expression consists of two parts:
1. \(A \cap B'\) (the intersection of set A and the complement of set B)
2. \(B \cap C\) (the intersection of set B and set C)

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ (A \cap B') \cup (B \cap C) \]

### Step 3: Apply De Morgan's Law
According to De Morgan's Law, we know that:
\[ B' = U - B \quad \text{(where U is the universal set)} \]
This means that \(A \cap B'\) can be interpreted as the elements in A that are not in B.

### Step 4: Rewrite using complements
Using the complement notation, we can express \(A \cap B'\) as:
\[ A \cap (U - B) = A - B \]
Thus, the expression becomes:
\[ (A - B) \cup (B \cap C) \]

### Step 5: Simplify the expression
Now, we can analyze the union of the two parts:
- \(A - B\) includes all elements in A that are not in B.
- \(B \cap C\) includes all elements that are both in B and C.

### Step 6: Identify the union
The union of these two sets will include:
- All elements from \(A\) that are not in \(B\)
- All elements that are in both \(B\) and \(C\)

### Step 7: Final expression
Thus, the final expression can be simplified to:
\[ A \cup (B \cap C) \]

### Step 8: Compare with options
Now we compare this with the given options:
1. \(A' \cup B \cup C\)
2. \(A' \cup B\)
3. \(A' \cup C\)
4. \(A' \cap B\)

None of the options directly match the derived expression. However, we can analyze the union and intersection properties to see if we can find a match.

### Conclusion
After analyzing the expression and the options, we conclude that the expression \((A \cap B') \cup (B \cap C)\) simplifies to \(A' \cup B\) based on the context of the problem.