1. Here we consider sets A, B, C U where U is a universal set.

(a) If A B determine the relationship between A^c and B^c .

(b) If AB C determine the relationship between A \ C and A B.

(c) Determine P(A) P(A^c ).

(d) Is P(A) P(A^c) = P(U)?

(e) Is P(A) \ P(B) P(A r B)?

2. Express the following statements in the symbolic language using symbols as instructed in Problem group 1 and then negate the statement. Use only symbols =, <, and arithmetic operations like +, , and similar. In (e) consider a set A Q.

(a) Every positive integer can be expressed as a sum of three squares of integers.

(b) For every real number r the equation 8x + r = 0 has precisely one real solution.

(c) There are at least three different integers in the interval (2, 2).

(d) For every real number x larger than 1 the value 1/x is smaller than 1 but larger than 0.

(e) Between any two distinct integers there is an element of A.