QUESTION 3 Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. Give an example to help you prove each truth value I. \(\forall x \exists y(x + y = 1)\) II. \(\exists x \exists y((x + 2y = 2) \land (2x + 4y = 5))\) III. \(\forall x \forall y \exists z(z = \frac{x + y}{2})\)
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Step 1
The statement ∃x∀y(x+y=1) is true. To prove this, we can choose x=0. For any value of y, when we add it to x, we get 0+y=y, which equals 1 when y=1. Show more…
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Vincenzo Z.
Determine the truth values of the following statements, if the domain for all variables is the set of real numbers: a) ∀x∀y(x^2 = y^2 → x = y). b) ∀x∃y(x = y^2). c) ∀x(x ≠ 0 → ∃y(xy^3 = 1)). d) ∃x∀y(y ≠ 0 → xy = 1). e) ∀x∀y∃z(z^2 = x^2 + y^2).
Drew S.
Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \forall x \exists y\left(x^{2}=y\right)} & {\text { b) } \forall x \exists y\left(x=y^{2}\right)} \\ {\text { c) } \exists x \forall y(x y=0)} & {\text { d) } \exists x \exists y(x+y \neq y+x)}\end{array} $$ $$ \begin{array}{l}{\text { e) } \forall x(x \neq 0 \rightarrow \exists y(x y=1))} \\ {\text { f) } \exists x \forall y(y \neq 0 \rightarrow x y=1)} \\ {\text { g) } \forall x \exists y(x+y=1)} \\ {\text { h) } \exists x \exists y(x+2 y=2 \wedge 2 x+4 y=5)} \\ {\text { i) } \forall x \exists y(x+y=2 \wedge 2 x-y=1)} \\ {\text { j) } \forall x \forall y \exists z(z=(x+y) / 2)}\end{array} $$
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