# Translating a statement to logic.

1. Jan 28, 2012

### lizarton

1. The problem statement, all variables and given/known data

Translate the following sentences into propositional or predicate logic. Use the shorthand
symbols (e.g. $\vee$) and define the meaning of each of your predicates and propositional variables. Be sure to include a domain (aka replacement set) for each quantified variable. You may need to rewrite the sentences slightly so as to make a variable more explicit.

"Irrational numbers have decimal expansions that neither terminate nor become
periodic".

2. Relevant equations

3. The attempt at a solution

Since the predicate refers to the decimal expansion of irrational numbers, I don't know whether to (a) declare/quantify a variable that represents the decimal expansion of an irrational number, or to (b) declare/quantify a variable that represents an irrational number.

(a)
T(x) is "d terminates".
P(x) is "d becomes periodic".
D is the set of all decimal expansions of irrational numbers.

$\left(\forall d \in D\right)\left(\neg T(d) \wedge \neg P(d)\right)$

...versus:
(b)
D(x) is “x has a decimal expansion”.
T(D(x)) is “the decimal expansion of x terminates”.
P(D(x)) is “the decimal expansion of x becomes periodic”.
I is the set of all irrational numbers.

$\left(\forall x \in I\right)\left( D(x) \wedge \neg T(D(x)) \wedge \neg P(D(x)) \right)$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution