p : xy =yx , is true for every real number x and y
q : There exists real number x and y for which xy=yx is not true. above pair of statement are

Options

A. Negation of each other
B. Not negation of each other
C. Converse of each other
D. Contrapostive of each other

Answer

To determine the relationship between the statements \( P \) and \( Q \), we need to analyze them carefully.

### Step 1: Understand the Statements

- **Statement \( P \)**: \( xy = yx \) is true for every real number \( x \) and \( y \).
- **Statement \( Q \)**: There exists real numbers \( x \) and \( y \) for which \( xy \neq yx \).

### Step 2: Analyze Statement \( P \)

The statement \( P \) asserts that the multiplication of any two real numbers \( x \) and \( y \) is commutative. This means that for any real numbers, the order in which they are multiplied does not affect the result.

### Step 3: Analyze Statement \( Q \)

The statement \( Q \) claims that there are some real numbers \( x \) and \( y \) such that the multiplication is not commutative, i.e., \( xy \neq yx \).

### Step 4: Determine the Relationship

- If \( P \) is true (i.e., multiplication is commutative for all real numbers), then \( Q \) must be false (there are no real numbers for which multiplication is not commutative).
- Conversely, if \( Q \) is true (there exist some real numbers for which multiplication is not commutative), then \( P \) must be false (multiplication is not commutative for all real numbers).

This establishes that \( P \) and \( Q \) are negations of each other.

### Conclusion

Since \( P \) and \( Q \) are negations of each other, the correct answer is that they are negation of each other.

### Final Answer

The pair of statements \( P \) and \( Q \) are negation of each other.

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