10. Suppose x and y indicate particular real numbers. Write conditions that express the following, by using compar- isons (such as x > 0) and the basic operations of logic. (d) Neither x nor y is positive. 24. By quoting the parts of Theorem 2, verify the follow- ing logical equivalences. In each case, start from the left side and use parts of the theorem to change the prob- lem, ending up with the right side. (See Example 15 and the solution to Practice Problem 9.) (b) (p v t)^(p Vc) = p 12. For each of the following English sentences, specify a domain and a predicate, and write the quantified state- ment symbolically using either form discussed in Exer- cise 11. (a) Even numbers are never prime. (c) There are no integers a and b for which a = 2. 14. Write the negation of each of the following statements, using Proposition 1 and the rules for negating propo- sitions to simplify each to the point that no - symbol occurs to the left of a quantifier. (Recall that Z denotes the set of all integers, and R denotes the set of all real numbers.) (c) Vx e Z, 3y e R, X = 2y y 15. For each statement in Exercise 14, is the original state- ment or its negation true? (c) Vx e Z, 3y e R, X = 2y 10. Let D be the set {1, 3, 5, 7, 8, 10, 11, 12}. For each of the following, decide whether it is true for all the elements of D. If it is not, give a counterexample. (b) If x is odd, then x > 7. (e) If x > 12, then x is even. 14. Express each of the following statements using predi- cates and the quantifier V: (d) For every positive real number x, if x V2