# Discrete Mathematics Lecture 2 9/3/2015 1

```Discrete Mathematics
Lecture 2
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9/3/2015
Introduction
 What is Discrete Mathematics? Discrete Mathematics
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concerns processes that consist of a sequence of individual
steps.
LOGIC:
Logic is the study of the principles and methods that
distinguish between a valid and an invalid argument.
SIMPLE STATEMENT:
A statement is a declarative sentence that is either true or
false but not both. A statement is also referred to as a
proposition
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1 Logic Concepts
 The logic imitate our intuitions feelings by setting down
explaining ideas that behave analogously
 The logic furnishes statements that describe the surrounding
world that can be true/false.
 Used in numerous applications: circuit design, programs,
verification of correctness of programs, artificial intelligence,
etc.
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Propositional Logic
• Is a declarative statement that is either true or false, but not both.
It is not an Opinion, a Command, or a Question. Such as:
 – Washington, D.C., is the capital of USA
 – California is adjacent to New York
 – 1+1=2
 – 2+2=5
• Not declarative
 –What time is it?
Question
Command
 It is a nice day
opinion
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Propositional Logic
• Elementary/Simple statement is atomic, with a verb, subject
and object but no connectives (NOT, OR, AND, If…Then,
Iff)
• Construct correct mathematical arguments.
• Give precise meaning to mathematical statements
• Focuses on the relationships among statements.
• Rules are used to distinguish between valid (True) and invalid
(False) arguments/statement. Hence logic assures us for any
combination/connection of these statements as being
True/False.
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Examples
 Propositions
 1) Grass is green.
 2) 4 + 2 = 6
 3) 4 + 2 = 7
 Not Propositions
 1) Close the door.
 2) x is greater than 2.
 3) He is very rich
 4) There are four fingers in a
hand.
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Examples
 Rule: If the sentence is preceded by other sentences that
make the pronoun or variable reference clear, then the
sentence is a statement.
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UNDERSTANDING STATEMENTS
 1) x + 2 is positive.
 2) May I come in?
 3) Logic is interesting.
 4) It is hot today.
 5) -1 &gt; 0
 6) x + y = 12
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Not a statement
Not a statement
A statement
A statement
A statement
Not a statement
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False, True, Statements
 Axiom : False is the opposite to Truth.
 A statement/declaration/declarative-statement is a
 description of something (sentence), which Is Not an
Opinion, a Command, or Question.
 A statement Is A Sentence That Is Either True Or False,
but not both simultaneously.
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Examples
 Examples of statements:
 – I’m 35 years old.
 – I have 100 children.
 – I always tell the truth.
 – I’m lying to you.
 Q’s: Which statements are: True? False? Both? Neither?
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EXAMPLES:
 a. 2+2 = 4,
 b. It is Sunday today
 If a proposition is true, we say that it has a truth value of
&quot;true”. If a proposition is false, its truth value is &quot;false&quot;.
The truth values “true” and “false” are, respectively,
denoted by the letters T and F.
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Logical connectives
 EXAMPLES:
 1. “3 + 2 = 5” and “Makka is a city in KSA”
 2. “The grass is green” or “ It is hot today”
 3. “Discrete Mathematics is not difficult to me” .
 AND, OR, NOT are called LOGICAL CONNECTIVES.
 SYMBOLIC REPRESENTATION
 Statements are symbolically represented by letters such as p, q,
r,...
 EXAMPLES:
 p = “Riyadh is the capital of KSA”
 q = “17 is divisible by 3”
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Symbolic logic
 In symbolic logic, we use lower case letters such as p, q, r,
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and s to represent statements. Here are two examples:
p: Riyadh is the capital of KSA.
q: Abin Al-Haythim is one of the greatest muslim scientists.
Hence:
The letter p represents the 1st statement.
The letter q represents the 2nd statement.
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Logical operators /connevtives
 • If you're wealthy or well educated, then you'll be happy.
 We can break this statement down into three basic sentences:
 You're wealthy,You're well educated,You'll be happy. These
sentences are called simple statements because each one
conveys/takes/sends/carries/transfers/delivers one idea
with no connecting words. Statements formed by combining
two or more simple statements are called compound
statements. Logical connectives are used to join simple
statements to form a compound statement. These
Connectives include the words AND, OR, IF . . . THEN, and
IF AND ONLY IF.
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Compound statements
 Compound statements appear throughout written and
spoken language. We need to be able to understand the logic
of such statements to analyze information objectively.
 we will concentrate our analysis on four kinds of compound
statements.
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TABLE
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CONNECTIVE
MEANING
SYMBOLE
CALLED
Negation
Not
&not;
Tilde
Conjunction
And
∧
Hat
Disjunction
Or
∨
Vel
Conditional
If …..then ….
→
Arrow
Biconditional
If and only if
↔
Double arrow
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Exercise sets
 determine whether or not each sentence is a
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Statement. Specify which is True and which is False.
1. George W. Bush was the Democratic candidate for
president in 2004.
2. John Kerry was the Republican candidate for president in
2004.
3. Take the most interesting classes you can find.
4. Don't try to study on a Friday night in the dorms.
5. The average human brain contains 100 billion neurons.
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Exercise
 6. There are 2.500.000 rivets in the Eiffel Tower.
 7. Is the unexamined life worth living?
 8. Is this the best of all possible worlds?
 9. Some Catholic countries have legalized same-sex marriage.
 l0. Some U.S. presidents were assassinated.
 11. 9 + 6 = 16
 12 . 9 x 6 = 64
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