Which of the following is not true ?

Options

A. `A - (B uu C)= (A-B) nn (A - C)`
B. `(A uu B uu C)' = A'nn B' nn C'`
C. `A - (B - C) = (A-B) -C`
D. `(A Delta B) Delta C = A Delta (B Delta C)`

Explanation

To determine which of the given statements is not true, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Statement:** A - B ∪ C = (A - B) ∩ (A - C)

**Explanation:**
- A - B represents the elements in A that are not in B.
- C is a set that we are taking the union with.
- The right side, (A - B) ∩ (A - C), represents the elements that are in A but not in B and also not in C.

**Conclusion:** This statement is true.

### Step 2: Analyze Option 2
**Statement:** (A ∪ B ∪ C)' = A' ∩ B' ∩ C'

**Explanation:**
- This statement is based on De Morgan's theorem, which states that the complement of the union of sets is equal to the intersection of their complements.
- Therefore, taking the complement of A ∪ B ∪ C gives us A' ∩ B' ∩ C'.

**Conclusion:** This statement is true.

### Step 3: Analyze Option 3
**Statement:** A - (B - C) = A - B - C

**Explanation:**
- The left side, A - (B - C), means we take the elements in A and remove those that are in B but not in C.
- The right side, A - B - C, means we take elements in A and remove those in B and then remove those in C.
- These two expressions are not equivalent because removing B and then C is not the same as removing only B's elements that are not in C.

**Conclusion:** This statement is not true.

### Step 4: Analyze Option 4
**Statement:** A Δ B Δ C = (A Δ B) Δ C

**Explanation:**
- The symbol Δ represents the symmetric difference, which is defined as the elements that are in either of the sets but not in their intersection.
- The associative property holds for symmetric difference, meaning that the grouping of sets does not affect the outcome.

**Conclusion:** This statement is true.

### Final Conclusion
After analyzing all options, we find that the statement in **Option 3** is not true.

### Answer:
The option that is not true is **Option 3: A - (B - C) = A - B - C**.

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