Discrete Math 18 Suppose that the domain of the propositiona

(Discrete Math)

18. Suppose that the domain of the propositional function P (x) consists of the integers 2, 1, 0, 1, and 2. Write out each of these propositions using disjunctions, conjunctions, and negations.

a) xP (x)

b) xP (x)

c) x¬P (x)

e) ¬xP (x)

Solution

a) x P(x), So it says that there exists an x for which P(x) is true.

Then P(2)P(1)P(0)P(1)P(2)

b) x P(x), Here it says that P(x) is true for every x.

Then P(2)P(1)P(0)P(1)P(2)

c) x ¬P(x),

Here it says \"There is at least one x for which P(x) is not true.\" This is the inverse of part B, which says \"P(x) is true for all x.\"

Therefore part C is

¬(P(2)P(1)P(0)P(1)P(2))

This can also be written as

¬P(2)¬P(1)¬P(0)¬P(1)¬P(2)

These two statements are equivalent, and are both correct.

e) ¬xP (x), here it says \"There exists no value of x for which P(x) is true\".

¬(P(2)P(1)P(0)P(1)P(2))