Question

Suppose the variable x represents children and the variable y represents treats, and: S(y): y is a sweet treat R(y): yis a salty treat, K(x):x is a child who is 10 or younger, T(x):x is a teenager and P(x,y):x likes y Translate into words: a) Vy(S(y)). b) 3xVy((K(x) A P(x, y)) c) 3y -(S(y) v R(y)) d) Express using quantifiers: Some children don't like salty treats. e) Express using quantifiers: All treats are either sweet or salty.

Answer 1

(a)

-Vy(S(y)) means that it is not the case that for all treats y, y is a sweet treat.

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Hence, the translation is:

"Not all treats are sweet."

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(b)

$\\* \exists x\forall y((K(x)\land P(x,y))) \text{ means that }\\*\text{there exists a child x such that for all treats y,} \\*\text{x is a child who is 10 or younger and x likes y.}$

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Hence, the translation is:

"There is a child who is 10 or younger who likes all treats."

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(c)

$\\* \exists y\lnot(S(y)\lor R(y)) \text{ means that } \\*\text{there exists a treat y such that it is is not the case that } \\*\text{y is a sweet treat or y is a salty treat.}$

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Hence, the translation is:

"There exists a treat which is neither sweet nor salty."

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(d)

"Some children don't like salty treats." can be rewritten as "There exists a child x such that for all treats y, if y is salty, then x does not like y."

ANSWER:

$\exists x\forall y(R(y)\to\lnot P(x,y))$

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(e)

"All treats are either sweet or salty." can be rewritten as "For all treats y, y is sweet or y is salty."

ANSWER:

Vy(S(y) V R(y))