Negation of `''AuuB=B` is the sufficient condition for `AsubeB''` is

Options

A. `AuuB=BandAcancelsubeB`
B. `AuuBneB" but "AsubeB`
C. If `AcancelsubeB," then "AuuBneB`
D. `AnnB=BandAsubeB`

Answer

To find the negation of the statement "A ∪ B = B is the sufficient condition for A ⊆ B", we can follow these steps:

### Step 1: Understand the Original Statement
The original statement can be expressed as:
- If (A ∪ B = B) then (A ⊆ B).

This is a conditional statement where:
- Let P: A ∪ B = B
- Let Q: A ⊆ B

Thus, the statement can be written as:
- P → Q

### Step 2: Write the Negation of the Conditional Statement
The negation of a conditional statement P → Q is given by:
- ¬(P → Q) which is equivalent to P ∧ ¬Q.

### Step 3: Identify P and Q
From our definitions:
- P is "A ∪ B = B"
- Q is "A ⊆ B"

### Step 4: Write the Negation
Now, we need to express the negation:
- ¬Q means "A is not a subset of B", which can be written as "A ⊈ B".

Therefore, the negation of the original statement becomes:
- P ∧ ¬Q
- This translates to: (A ∪ B = B) ∧ (A ⊈ B).

### Final Answer
The negation of "A ∪ B = B is the sufficient condition for A ⊆ B" is:
- (A ∪ B = B) ∧ (A ⊈ B).

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