Are that following pair of statements negation of each other?
` (i) `The relation `xy=yx` is true for every real number ` x` and `y` . .
`(ii)` There exists real number `x` and `y` for which the relation `xy=yx` is not true .

Answer

To determine whether the given statements are negations of each other, we will analyze both statements step by step.

### Step 1: Understand the First Statement
The first statement is:
- (i) "The relation \( xy = yx \) is true for every real number \( x \) and \( y \)."

This means that for all pairs of real numbers \( x \) and \( y \), the equation \( xy = yx \) holds true.

### Step 2: Negate the First Statement
To negate the first statement, we need to express that it is not true for all real numbers. The negation of "for every" (or "for all") is "there exists" (or "there is at least one"). Therefore, the negation of statement (i) is:
- "There exists real numbers \( x \) and \( y \) such that \( xy \neq yx \)."

### Step 3: Understand the Second Statement
The second statement is:
- (ii) "There exists real number \( x \) and \( y \) for which the relation \( xy \neq yx \) is true."

This statement explicitly states that there are at least some real numbers \( x \) and \( y \) for which the equation does not hold.

### Step 4: Compare the Two Statements
Now we compare the negation of the first statement with the second statement:
- The negation of (i) is: "There exists real numbers \( x \) and \( y \) such that \( xy \neq yx \)."
- Statement (ii) is: "There exists real number \( x \) and \( y \) for which the relation \( xy \neq yx \) is true."

Both statements express the same idea: that there are some real numbers for which the relation \( xy = yx \) does not hold.

### Conclusion
Since the negation of the first statement is identical to the second statement, we conclude that:
- Therefore, the given statements are negations of each other.

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