snhu Directions: Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit. PROBLEM 1 (a) The domain for all variables in the expressions below is the set of real numbers. Determine whether each statement is true or false. (i) Vx Jy (x + y > 0) The statement Valy(x + y ? 0) means that for any real number x, there exists another real number y such that when you add x and y together, the result is greater than or equal to zero. To confirm its truth: Consider any real number x. If x is positive, we can choose y = - x. This choice ensures that when we add x and y together, the result is x - x = 0, which is greater than or equal to zero. Similarly, if x is negative, we can still choose y = - x, resulting in x - x = 0, which meets the condition. Even if x is zero, we can choose any y (for example, y = 0), and the sum x + y will still be 0 ? 0. In every case, we can find a suitable y for any given x such that x + y ? 0. Therefore, the statement is true. (ii) Ex Vy (x . y > 0) The statement 3xVy(x * y > 0) asserts that there exists a real number x such that for all real numbers y, the product of x and y is greater than zero. If y = 0, then for any x (positive, negative, or zero), x * 0 = 0, which is not greater than zero. So the statement is false. (b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1.
ExTy( < 1) snhu (ii) The reciprocal of every positive number is also positive. Vx (x > 0 > > > 0)
snhu PROBLEM 2 Prove the following using the specified technique: (a) Let x and y be two real numbers such that x + y is rational. Prove by contrapositive that if x is irrational, then x - y is irrational. To prove by contrapositive, we start by assuming the negation of the consequent and the antecedent of the conditional statement, and then prove the negation of the antecedent. Given: x + y is rational. We want to prove: If x is irrational, then x - y is irrational. The contrapositive of the statement "If A, then B" is "If not B, then not A." Assume that x - y is rational. We'll show that if x is irrational, then x + y is irrational. Proof by Contradiction: Assume x is irrational but x - y is rational. If x - y is rational, then (x - y) + y = x is also rational because a rational number plus a rational number is