Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

0.2. Mathematical Statements 21
12. Let P(x) be the predicate, “3x + 1 is even.”
(a) Is P(5) true or false?
(b) What, if anything, can you conclude about ∃xP(x) from the truth
value of P(5)?
(c) What, if anything, can you conclude about ∀xP(x) from the truth
value of P(5)?
13. Let P(x) be the predicate, “4x + 1 is even.”
(a) Is P(5) true or false?
(b) What, if anything, can you conclude about ∃xP(x) from the truth
value of P(5)?
(c) What, if anything, can you conclude about ∀xP(x) from the truth
value of P(5)?
14. For a given predicate P(x), you might believe that the statements
∀xP(x) or ∃xP(x) are either true or false. How would you decide if
you were correct in each case? You have four choices: you could give
an example of an element n in the domain for which P(n) is true or
for which P(n) if false, or you could argue that no matter what n is,
P(n) is true or is false.
(a) What would you need to do to prove ∀xP(x) is true?
(b) What would you need to do to prove ∀xP(x) is false?
(c) What would you need to do to prove ∃xP(x) is true?
(d) What would you need to do to prove ∃xP(x) is false?
15. Suppose P(x, y) is some binary predicate defined on a very small
domain of discourse: just the integers 1, 2, 3, and 4. For each of the 16
pairs of these numbers, P(x, y) is either true or false, according to the
following table (x values are rows, y values are columns).
1 2 3 4
1 T F F F
2 F T T F
3 T T T T
4 F F F F
For example, P(1, 3) is false, as indicated by the F in the first row,
third column.
Use the table to decide whether the following statements are true
or false.
(a) ∀x∃yP(x, y).