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🚧 Fin 3.3

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tomit4 2026-06-05 23:42:01 -07:00
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@ -77,19 +77,27 @@ Page 165
the property that the $x$ the person gave you together with the _______ you the property that the $x$ the person gave you together with the _______ you
subsequently found satisfy _______. subsequently found satisfy _______.
$y \in E$; $y$; $P(x, y)$
2. To establish the truth of a statement of the form 2. To establish the truth of a statement of the form
"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$," "$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$,"
you need to find _______ so that no matter what _______ a person might you need to find _______ so that no matter what _______ a person might
subsequently give you, _______ will be true. subsequently give you, _______ will be true.
$x \in D$; $y \in E$; $P(x, y)$
3. Consider the statement 3. Consider the statement
"$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$." "$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
A negation for this statement is "_______." A negation for this statement is "_______."
"$\exists x$ such that $\forall y$, the property $P(x, y)$ is false."
4. Consider the statement 4. Consider the statement
"$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$." "$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
A negation for this statement is "_______." A negation for this statement is "_______."
"$\forall x, \exists y$ such that the property $P(x, y)$ is false."
5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the 5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the
statement statement
"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$" "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$"
@ -101,3 +109,6 @@ a. is true.
b. is false. b. is false.
c. may be true or may be false. c. may be true or may be false.
c is the answer, it may be true or false depending on the nature of the property
involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states.