Bush Willis

03/12/2024 · High School

Exercises 1. In each of the following, two open statements \( P(x, y) \) and \( Q(x, y) \) are given, where the domain of both \( x \) and \( y \) is \( Z \). Determine the truth value of \( P(x, y) \Rightarrow Q(x, y) \) for the given values of \( x \) and \( y \). a. \( P(x, y): x^{2}-y^{2}=0 \). and \( Q(x, y): x=y .(x, y) \in\{(1,-1),(3,4),(5,5)\} \). b. \( P(x, y):|x|=|y| \). and \( Q(x, y): x=y .(x, y) \in\{(1,2),(2,-2),(6,6)\} \). c. \( P(x, y): x^{2}+y^{2}=1 \). and \( Q(x, y): x+y=1 \). \( (x, y) \in\{(1,-1),(-3,4),(0,-1),(1,0)\} \). 2. Let \( O \) denote the set of odd integers and let \( P(x): x^{2}+1 \) is even, and \( Q(x): x^{2} \) is even. be open statements over the domain \( O \). State \( (\forall x \in O) P(x) \) and \( (3 y \in O) Q(x) \) in words. 3. State the negation of the following quantified statements.