Math 166
Logic – L.1, L.2
Texas A&M – Spring 2016
Chapter : Logic
L.1
Logic
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is the science of correct reasoning and of making valid conclusions.
Every concept must be clearly defined.
Every declarative sentence should be either True, or False, but not both!
Such declarative sentences are called STATEMENTS.
Commands, questions, exclamations, and ambiguous sentences cannot be statements.
Example 1 Which of the following sentences are statements?
a) Two times three equals six.
b) Do you have any money?
c) It's a beautiful day!
d) 1 + 1 = 5
e) Sit down.
f) x + y = 3
Simple and Compound Statements (Mark 'S' for Simple , 'C' for Compound)
Examples
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I love to read.
I love to read and I have a library.
I could go to the store or I could take a nap.
My dog does not like the thunder.
Do not talk in class.
Connectives
• In the above examples, 'and', 'or', and 'not' are known as connectives.
• A connective is a word or words that combines two or more simple statements into a
compound statement.
• Rules of Logic enable us to determine if the compound (not simple) statement is True or
False.
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Math 166
Logic – L.1, L.2
Texas A&M – Spring 2016
Different Kinds of Connectives
Let p and q be two statements where p: The board is white
q : The pen is red.
1. Conjunction (AND)
• is a statement of the form “p and q”.
• It's symbolically written as __________ , where p and q are two simple statements.
• p Λ q is true if BOTH p and q are true, else it's false.
2. Disjunction (OR)
• is a statement of the form “p or q”.
• It's symbolically written as __________ , where p and q are two simple statements.
• p V q is true if EITHER p or q or both is/are true. It's false if BOTH p and q are false
3. Exclusive Disjunction
• is symbolically written as ___________, where p and q are two simple statements.
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_________ is true if EITHER p is true or q is true, but not both.
Note : Unless mentioned otherwise, we will be referring to Inclusive OR every time we talk
about disjunction.
4. Negation (NOT)
• is a statement of the form “not p”, where p and q are two simple statements.
• It is written as ________
• ~p is true if 'p' is false, and vice verse.
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Math 166
Logic – L.1, L.2
Texas A&M – Spring 2016
Order of precedence if the logical connectives: ~, Λ, V
Example 2
Let p, q and r be the following statements:
p : Austin is the capital of Texas.
q: The dog is wet.
r: Area of the square is 16 square units.
I ) Translate the compound statement into symbolic form
a) Austin is the capital of Texas and the dog is wet.
b) Either Austin is the capital of Texas and the dog is not wet or the dog is wet and the square has
an area of 16 square units..
c) Austin is the capital of Texas and the area of the square is 16 square units, but the dog is not wet.
d) It is not true that Austin is the capital of Texas and the dog is wet.
II ) Write the statements corresponding to the following
a) p V q
b) (~ p V q ) Λ r
c) ~q V ~r
d) p V q
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Math 166
Logic – L.1, L.2
Texas A&M – Spring 2016
Example 3
Let p and q be two statements such that p is true and q is false, and you are given
p: The board is white
q : The pen is red.
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Math 166
Logic – L.1, L.2
Texas A&M – Spring 2016
Truth Tables
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The truth value of a statement is either true or false.
Rules of Logic enable us to determine if the (truth value of the) compound (not simple)
statement is True or False.
Example 4 Determine the truth value of the following statements, and state if they are a tautology, a
contradiction, or neither.
a) ~ p Λ ( p V q )
b) p Λ (~ p V q )
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Math 166
Logic – L.1, L.2
c) p Λ ~ p
d) p V ~ p
e) (p Λ q) Λ [(r V ~ p) Λ q]
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Texas A&M – Spring 2016