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Vocabulary Words logic proposition truth value compound proposition logical operation conjunction truth table disjunction exclusive or inclusive or negation
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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
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