Real Numbers

The given statement is "For all real numbers x and y, x + y = y + x".

The negation of "For all" is "There exists some" or "For some".

The negation of "=" is " ".

Therefore, the negation of the statement is: "For some real numbers x and y, x + y y + x".

Thus, the correct option is (C) for some real numbers x and y, x + y y + x.

The correct answer is: (B):

We are given that , , and are real numbers such that and . We are tasked with determining the relationship between and .

Step 1: Understanding the implications of and

We are given that , so initially, it would seem that should be less than . However, since , we must consider how dividing by a negative number affects the inequality.

Step 2: Dividing by a negative number

When you divide both sides of an inequality by a negative number, the direction of the inequality reverses. So, dividing by the negative number gives:

Step 3: Conclusion

Thus, since , the inequality reverses, and we have , which means the correct answer is (B): .

Given Equation: , where is the greatest integer function.

Step 1: Solve the quadratic equation for .

Let . The equation becomes:

Factorizing:

Solutions: or

Step 2: Translate back to .

Case 1: ⇒

Case 2: ⇒

Step 3: Combine the intervals.

The solution is =

Option Analysis:

(A) - Partial solution (misses [2,3))

(B) - Includes all solutions (correct)

(C) - Partial solution (misses [3,4))

(D) - Incorrect boundaries

Final Answer: