Let A, B, C be sets. Prove that if C ⊆A, then A −(B −C) = (A −B) ∪C. Analogously, for any real numbers a, b, and c, recall that a −(b −c) = (a −b) + c.

Let a, b be positive real numbers. Prove that a +b< v2(a² +b²).

If a and b are odd positive integers, and the sum of the integers, less than a and greater than b, is 1000; then find a and b.

2. Use prove by contrapositive to show that: (For real numbers a, b, c) If (a – b)² + (b – c)² + (c- a)? = 0, then a = b = c.

Determine whether the statement is true or false. Prove the statement if it is true, and give a counterexample if it is false. (i) For all real numbers x and y, [x – y] = [x] – [y]. (ii) Let a, b and c be integers. If a|(bc), then a|b or a|c. (iii) Let m and n be integers. If m is even and n is odd, then m² – n² is odd.

Let a and b be real numbers. Prove that if 4a+16b= 123, then a and b are not integers