If A, B and C Are subsets of a given set, then which one of the following isrelations is not correct?

Options

A. A. `A cup (A cap B) = A cup B`
B. B. `A cap (A cup B) = A`
C. C. `(A cap B) cup C= (A cup C) cap (B cup C)`
D. D. `(A cup B) cap C =(A cap C) cup (B cap C)`

Answer

To solve the problem, we need to analyze the relations involving the subsets A, B, and C of a given set and determine which one is not correct. Let's go through the options step by step.

### Step 1: Analyze Option A
We start with the relation:
\[ A \cup (A \cap B) \]

Using the distributive property of sets, we can rewrite this as:
\[ A \cup (A \cap B) = A \cup (A \cap B) \]

By the property of sets, we know that:
\[ A \cup (A \cap B) = A \]

Thus, the expression simplifies to:
\[ A \]

Now, let's compare this with the relation given in Option A:
\[ A \cup B \]

Since:
\[ A \neq A \cup B \]
(assuming B is not empty), we conclude that:
\[ A \cup (A \cap B) \neq A \cup B \]

**Conclusion for Option A**: This relation is incorrect.

### Step 2: Analyze Option B
Now, we check the relation:
\[ A \cap (A \cup B) \]

Using the property of sets, we can rewrite this as:
\[ A \cap (A \cup B) = A \]

This is because the intersection of a set with its union with another set is always the original set.

**Conclusion for Option B**: This relation is correct.

### Step 3: Analyze Option C
Next, we analyze:
\[ A \cap (B \cup C) \]

Using the distributive property, we can express this as:
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]

This expression is valid based on the distributive law of sets.

**Conclusion for Option C**: This relation is correct.

### Step 4: Analyze Option D
Finally, we check:
\[ A \cup (B \cap C) \]

Using the distributive property, we can express this as:
\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \]

This expression is also valid based on the distributive law of sets.

**Conclusion for Option D**: This relation is correct.

### Final Conclusion
After analyzing all options, we find that:
- Option A is incorrect.
- Options B, C, and D are correct.

Thus, the answer to the question is:
**Option A is not correct.**

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