7.2 Predicates P and Q are defined below. The domain of discourse is the set of

Question

7.2
Predicates P and Q are defined below. The domain of discourse is the set of all positive integers.

    P(x): x is prime
    Q(x): x is a perfect square (i.e., x = y2, for some integer y)

Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value.
(d)
x (Q(x) P(x))

(e)
x (¬Q(x) P(x))

7.5
In the following question, the domain of discourse is the set of employees of a company. Define the following predicates:

    A(x): x is on the board of directors
    E(x): x earns more than $100,000
    W(x): x works more than 60 hours per week

Translate the following logical expressions into English:
(e)
x (E(x) (A(x) W(x)))

(f)
x (A(x) ¬E(x) W(x))

Explanation / Answer

d
x (Q(x) P(x))
there exist x, where x is prime AND x is perfect square

if x is a perfect square, there exist y belongs such that x = y*y. If x = y*y, x has a factor y, so x cannot be prime.

so x (Q(x) P(x)) -> False

e
x (¬Q(x) P(x))
for all x, x is not perfect square or x prime

let us consider x is a perfect square.
¬Q(x) = false
P(x) = false

So x (¬Q(x) P(x)) -> false

7.5
e
x (E(x) (A(x) W(x)))
for all x, if x earns more than $100,000 then x is on the board of directors or x works more than 60 hours per week

f
x (A(x) ¬E(x) W(x))
There is atleast a employee who is on the board of directors and x works more than 60 hours per week but does not earn more than $100,000

I hope this makes you happy!! :D

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