Propositions over a Universe

In document Applied Discrete Structures (Page 72-75)

3.6.1 Propositions over a Universe

Consider the sentence “He was a member of the Boston Red Sox.” There is no way that we can assign a truth value to this sentence unless “he” is specified. For that reason, we would not consider it a proposition. However, “he” can be considered a variable that holds a place for any name. We might want to restrict the value of “he” to all names in the major-league baseball record books. If that is the case, we say that the sentence is a proposition over the set of major-league baseball players, past and present.

Definition 3.6.1 Proposition over a Universe. Let U be a nonempty set. A proposition overU is a sentence that contains a variable that can take on any value inU and that has a definite truth value as a result of any such

substitution. ♢

Example 3.6.2 Some propositions over a variety of universes. (a) A few propositions over the integers are4x23x= 0,0n5, and “k

is a multiple of 3.”

(b) A few propositions over the rational numbers are 4x23x= 0,y2 = 2,

and(s1)(s+ 1) =s21.

(c) A few propositions over the subsets ofP are(A=)(A=P), 3∈A, andA∩ {1,2,3} ̸=.

□ All of the laws of logic that we listed in Section 3.4 are valid for propositions over a universe. For example, ifpandqare propositions over the integers, we can be certain thatp∧q⇒p, because(p∧q)→pis a tautology and is true no matter what values the variables inpandqare given. If we specifypandq to bep(n) :n <4 andq(n) :n <8, we can also say thatpimpliesp∧q. This is not a usual implication, but for the propositions under discussion, it is true. One way of describing this situation in general is with truth sets.

3.6.2 Truth Sets

Definition 3.6.3 Truth Set. Ifpis a proposition overU, the truth set ofp

isTp={a∈U |p(a)is true}. ♢

Example 3.6.4 Truth Set Example. The truth set of the proposition

{1,2} ∩A = , taken as a proposition over the power set of {1,2,3,4} is

{∅,{3},{4},{3,4}}. □

Example 3.6.5 Truth sets depend on the universe. Over the universeZ (the integers), the truth set of4x23x= 0is{0}. If the universe is expanded

to the rational numbers, the truth set becomes{0,3/4}. The termsolution set is often used for the truth set of an equation such as the one in this example. □

CHAPTER 3. LOGIC 59 Definition 3.6.6 Tautologies and Contradictions over a Universe. A proposition overU is a tautology if its truth set is U. It is a contradiction if

its truth set is empty. ♢

Example 3.6.7 Tautology, Contradiction overQ. (s1)(s+ 1) =s21

is a tautology over the rational numbers.x22 = 0is a contradiction over the

rationals. □

The truth sets of compound propositions can be expressed in terms of the truth sets of simple propositions. For example, if a Tpq if and only if a

makesp∧qtrue. This is true if and only ifamakes bothpandqtrue, which, in turn, is true if and only ifa∈Tp∩Tq. This explains why the truth set of

the conjunction of two propositions equals the intersection of the truth sets of the two propositions. The following list summarizes the connection between compound and simple truth sets

Tpq =Tp∩Tq

Tp∨q =Tp∪Tq

T¬p=Tpc

Tp↔q= (Tp∩Tq)(Tpc∩Tqc)

Tpq =Tpc∪Tq

Table 3.6.8: Truth Sets of Compound Statements

Definition 3.6.9 Equivalence of propositions over a universe. Two propositions,pandq, are equivalent ifp↔qis a tautology. In terms of truth sets, this means thatpandqare equivalent ifTp=Tq . ♢

Example 3.6.10 Some pairs of equivalent propositions.

(a) n+ 4 = 9andn= 5are equivalent propositions over the integers. (b) A∩ {4} ̸= and4∈Aare equivalent propositions over the power set of

the natural numbers.

Definition 3.6.11 Implication for propositions over a universe. If p andqare propositions overU,pimpliesq ifp→qis a tautology. ♢ Since the truth set of p q is Tpc∪Tq, the Venn diagram for Tpq in

Figure 12shows thatp⇒qwhenTp⊆Tq.

CHAPTER 3. LOGIC 60 Example 3.6.13 Examples of Implications.

(a) Over the natural numbers: n 4 n 8 since {0,1,2,3,4} ⊆

{0,1,2,3,4,5,6,7,8}

(b) Over the power set of the integers: |Ac|= 1impliesA∩ {0,1} ̸=

(c) Over the power set of the integers,A⊆ even integers ⇒A∩odd integers =

3.6.3 Exercises for Section 3.6

1. IfU =P({1,2,3,4}), what are the truth sets of the following propositions? (a) A∩ {2,4}=.

(b) 3∈Aand1 /∈A. (c) A∪ {1}=A.

(d) Ais a proper subset of{2,3,4}. (e) |A|=|Ac|.

2. Over the universe of positive integers, define

p(n): nis prime andn <32. q(n): nis a power of 3. r(n): nis a divisor of 27.

Table 3.6.14

(a) What are the truth sets of these propositions?

(b) Which of the three propositions implies one of the others?

3. If U = {0,1,2}, how many propositions over U could you list without listing two that are equivalent?

4. Given the propositions over the natural numbers:

p:n <4, q: 2n >17, and r:nis a divisor of18 Table 3.6.15

What are the truth sets of: (a) q

(b) p∧q

(c) r (d) q→r

5. Suppose thatsis a proposition over{1,2, . . . ,8}. IfTs={1,3,5,7}, give

two examples of propositions that are equivalent tos. 6.

CHAPTER 3. LOGIC 61 itive integers:

p(n) :nis a perfect square andn <100 q(n) :n=|P(A)|for some setA. (b) DetermineTpq forpandqabove.

7. Let the universe beZ, the set of integers. Which of the following propo- sitions are equivalent overZ?

a: 0< n2<9 b: 0< n3<27

c: 0< n <3 Table 3.6.16

In document Applied Discrete Structures (Page 72-75)