Propositional Functions and Quantifiers

Determine the truth value of the each of these statements if the domain consists of all integers

1. $\forall n (n+1 > n)$
2. $\exists n (n^3 = -1)$
3. $\exists n (2n = 5n)$
4. $\forall n (3n \le 4n)$
5. $\forall x \forall y ((x^2 = y^2) \to (x = y))$

Determine the truth value of each of the following statements if the domain consists of all real numbers.

1. $\exists x(x^2 = 2)$
2. $\exists x (x^2 = -1)$
3. $\forall x (x^2 + 2 \ge 1)$
4. $\forall x (2x > 2)$
5. $\forall x (x+4)^2 = x^2 + 16$

Translate these statements into English where $F(x)$ is “$x$ is fast” and $A(x)$ is “$x$ is an athlete”, here the domain is the set of people.

1. $\forall x (A(x) \to F(x))$
2. $\forall x (F(x) \land A(x))$
3. $\exists x (F(x) \to A(x))$
4. $\exists x (A(x) \land \neg F(x))$

Let $C(x)$ denote the predicate “$x$ is in the correct place”, let $E(x)$ denote “$x$ is in excellent condition”, and let $T(x)$ denote “$x$ is a tool” where the domain of each predicate is the set of objects in a garage. Translate each into plain English:

1. $\exists x \neg C(x)$
2. $\exists x (T(x) \land \neg C(x))$
3. $\forall x (T(x) \to (C(x) \land E(x)))$

Simplify the statements below so that negations are only directly next to the predicates.

1. $\neg \exists x \forall y (\neg O(x) \vee E(y))$.
2. $\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))$.

3. There is a number $n$ for which no other number is either less than or equal to $n$.

4. It is false that for every number $n$ there are two other numbers which $n$ is between.

Consider the statement, “There is a building on the campus of some college in the United States in which every room is painted white.”

1. Express the statement using quantifiers. Be sure to define your predicate function and specify the domain of each of the three variables.
2. Express the negation of the above logical quantified statement so that no negation is to the left of a quantifier.
3. Write the negat…