John Rosson

Thursday February 8, 2007

Survey of Mathematical IdeasMath 100

Chapter 3, Logic

Introduction to Logic

1. Statements and Quantifiers

2. Truth Tables and Equivalent Statements

3. The Conditional

4. More on the Conditional

5. Analyzing Arguments Truth Tables

StatementsA statement is a declarative sentence that is either true or false but not both. Such a sentence is said to have a truth value of either true or false (but not both).

Examples of statements. What are their truth values?

Millersville University is in Millersville PA.

2+2=4

2+2=3

{a,b,c} {a,b}

240 =1099511627776

240 =1099511627775

Statements

Examples of nonstatements.

2+2

x+2=3

Logic is an interesting subject.

Students ought to study.

This sentence is false.

Your professor always lies.

Statements

Examples of compound statements. What are their truth values?

Millersville University is in Millersville PA and 2+2=3.

Millersville University is in Millersville PA or 2+2=3.

If x=1 then x+2=3.

If xx then 2+2=3.

Simple statements can be combined into compound statements using logical connectives such as and, or, not, if…then.

NegationsThe negation of a statement is indicated by words and phrases like not, it is not the case that, it is not true that, does not etc. For mathematical relations negation often indicated by a slash (/) through the relation.

Examples of statements. What are their truth values?

Millersville University is not in Millersville PA.

It is not the case that Millersville University is not in Millersville PA.

2+24

2+23

{a,b,c} {a,b}/

QuantifiersSentences that contain variables are usually not statements because the truth value depends on the value assigned the variable. These sentences are indeterminate or predicates.

x xx = x

x is a university in Millersville PA.

x+3=4

x {1,2,3,….}

x+23

Examples of indeterminate sentences. Give a value making the sentence true and one making it false.

Examples of statements. What are their truth values?

QuantifiersQuantifiers are logical elements that give truth values to indeterminate sentences. Universal quantifiers are words like all, each, every and none. Existential quantifiers are words and phrases like some, there exists and (for) at least one.

All universities are in Millersville PA.

Some universities are in Millersville PA.

No universities are in Millersville PA.

For at least one value of x, x+3=4.

For all values of x, x {1,2,3,….}.

For some values of x, x+23

Examples of quantified sentences. Give the truth values.

SymbolsTo simplify work with logic and to bring out relations between logical (or deductive) reasoning and algebraic calculation the logical elements we have been talking about are replaced by symbols.

Logical Element Symbol Type

statements p, q, r, s,…. Value true or false

and Conjunction

or Disjunction

not Negation

If… then Conditional

For all Universal Quantifier

There exists Existential Quantifier

∧∨

∀∃

Symbols

Let p=“Millersville University is in Millersville PA” q=“2+2=3”

Examples of formalized statements.

Statement In symbols

Millersville University is in Millersville PA p

Millersville University is not in Millersville PA. p

Millersville University is in Millersville PA and 2+2=3. pq

Millersville University is in Millersville PA or 2+2=3. pq

It is not the case that Millersville University is not in Millersville PA. p

SymbolsFor indeterminate sentences, we indicate the variable in parentheses.Let p(x)=“x is a university in Millersville PA.” q(x)=“x+3=4” r(x)= “x {1,2,3,….}”

Examples of formalized statements.

Statement In symbols

All universities are in Millersville PA. x p(x)

Some universities are in Millersville PA. x p(x)

No universities are in Millersville PA. x p(x)

For at least one value of x, x+3=4. x q(x)

For all values of x, x {1,2,3,….}. x r(x)

Set Theory Reprise

Let us reconsider the definition of the set theory operations using logic symbols.

€

A∩ B ={x | x ∈ A∧x ∈ B}

A∪B ={x | x ∈ A∨x ∈ B}

A − B ={x | x ∈ A∧~ (x ∈ B)}

Symbolic LogicIt follows from this symbolization process, some simple logical facts and the fact that everything in mathematics is a set that all statements in mathematics (true or false) can be constructed as lists of the following symbols:

x, |, , =, , ,

In other words, mathematics is a subset of the natural numbers, base 7. Therefore, mathematics itself has cardinality 0.

How then can mathematics talk about the the set real numbers, which we have seen has cardinality c > 0?

Here is what “It is not true that for some x, xx” would look like: ~x|~=x|x|

Believe it or not someone actually tried to write out all of mathematics using (essentially) the above seven symbols. The result was the Principia Mathematica of Bertrand Russell and Alfred North Whitehead.

Assignments 3.2, 3.3, 3.4, 3.6

Read Section 3.2 Due February 13

Exercises p. 111 1-20 , 21-25, 45-55.

Read Sections 3.3, 3.4 Due February 15

Exercises p. 120 1-18, 55-63

Exercises p. 128 1-16, 19-38.

Read Section 3.5, 3.6 Due February 20

Exercises p. 145 1-23 odd, 27,29,47