Section 1.1

Propositions and Logical Operations

Introduction

• Remember that discrete is – the study of decision making in non-continuous

systems.

• That is, it is mathematics when there are a finite number of numbers.

Vocabulary Words

• logic• proposition• truth value• compound proposition• logical operation

• conjunction• truth table• disjunction• exclusive or• inclusive or• negation

In-Class Activity #1

• Get in groups of 3 or 4 students.• Each person should print their name at the

top of the paper I will hand out.• As a group, complete Part One

Let’s test the CRS

• Go to SocrativeEither in the app you downloaded or

through a student login on the website.

• Enter Room ITTC328

Which of these is the best definition of “Logic”

• A = Statements that are either true or false• B = The study of formal reasoning• C = The study of whether or not things

make sense• D = a set of rules

Logic

• “Logic is the study of formal reasoning”• (Mathematical logic is a tool for dealing

with formal reasoning)

Logic

• In a nutshell:– Logic does:

• Assess if an argument is valid/invalid– Logic does not directly:

• Assess the truth of atomic statements

What is the Difference

• Logic can deduce that:– Cedar Falls is in the USA

• given these facts:– Cedar Falls is in Iowa– Iowa is a part of the USA

• and the definitions of:– ‘to be a part of’– ‘to be in’

• But logic knows nothing of whether these facts actually hold in real life!

Logic

• An Argument is a sequence of statements aimed at demonstrating the truth of an assertion

• The assertion at the end of the sequence is called the Conclusion, and the preceding statements are called Premises.

Logic

• A Statement (also Proposition) is a sentence that is true or false but not both

• Definition #2 on your sheet– A statement that is either true of false (but not

both)

In-Class Activity #2

• With your group, complete Part Two on your sheet.

One at a time let’s see what you think.

1. 2 + 3 = 72. Julius Caesar was president of the United States.3. What time is it? 4. Be quiet ! 5. The difference of two primes. 6. 2 + 2 = 4 7. Washington D.C. is the capital of New York. 8. How are you?

Compound Propositions

• How did you define this in part one?– Connecting individual propositions with logical

operations.

Compound Propositions

• Most of the things we do in logic look at combinations of several propositions. For example:– My number is both an even number and a prime

number.– I am thinking of an odd number that is less than

20.

Logic

• In order to help us write “human language” based statements (I was going to type “English” but that’s probably not very accurate) in a more precise manner, we have come up with a variable-based and “mathematical” system for dealing with logic.

Logic

• To illustrate the logical form of arguments, we use letters of the alphabet (often p, q, and r) to represent the propositions of an argument.

Logic

• “My number is both an even number and a prime number.”

p represents “my number is even”q represents “my number is prime”

So we write:p and q

Logic Symbols

• There are a couple of base connectives that are common in logic. Each of these has been assigned a symbol.Not ¬And Or

• What are the “proper” names for these?

Truth Tables

• A truth table shows the relationship between various (often related) statements.

• It’s size depends on the number of independent variables represented in the statements– N independent atomic formulae (variables)

2N rows

Negationp ¬ pT

F

Conjuction (and)p q p q

T T

T F

F T

F F

Disjunction (or)p q p q

T T

T F

F T

F F

In Class Activity #3

• Complete Part Three with your team using the headers given below.

Activity

• Let– s = stocks are increasing– i = interest rates are steady

• How would we write– Stocks are increasing but interest rates are steady– Neither are stocks increasing nor are interest

rates steady

Activity

• Let– M = Juan is a math major– C = Juan is a computer science major

• How would we write– Juan is a math major but not a computer science

major