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

0.2. Mathematical Statements 17
and R(x) standing for “x is a rectangle”. The sentence we are looking at is,
S(x) → R(x).
This is neither true nor false, as it is not a statement. But come on! We all
know that we meant to consider the statement,
∀x(S(x) → R(x)),
and this is what our convention tells us to consider.
Similarly, we will often be a bit sloppy about the distinction between
a predicate and a statement. For example, we might write, let P(n) be the
statement, “n is prime,” which is technically incorrect. It is implicit that
we mean that we are defining P(n) to be a predicate, which for each n
becomes the statement, n is prime.
Exercises
1. For each sentence below, decide whether it is an atomic statement, a
molecular statement, or not a statement at all.
(a) Customers must wear shoes.
(b) The customers wore shoes.
(c) The customers wore shoes and they wore socks.
2. Classify each of the sentences below as an atomic statement, a molecular
statement, or not a statement at all. If the statement is molecular, say
what kind it is (conjunction, disjunction, conditional, biconditional,
negation).
(a) The sum of the first 100 odd positive integers.
(b) Everybody needs somebody sometime.
(c) The Broncos will win the Super Bowl or I’ll eat my hat.
(d) We can have donuts for dinner, but only if it rains.
(e) Every natural number greater than 1 is either prime or composite.
(f) This sentence is false.
3. Suppose P and Q are the statements: P: Jack passed math. Q: Jill
passed math.
(a) Translate “Jack and Jill both passed math” into symbols.
(b) Translate “If Jack passed math, then Jill did not” into symbols.
(c) Translate “P ∨ Q” into English.
(d) Translate “¬(P ∧ Q) → Q” into English.