1. Adapted from Discrete Math
Mathematical Logic
Adapted from Discrete Math

2. Learning Objectives Learn about sets
Explore various operations on sets
Become familiar with Venn diagrams
Learn how to represent sets in computer memory
Learn about statements (propositions)
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3. Learning Objectives
Learn how to use logical connectives to combine statements
Explore how to draw conclusions using various argument forms
Become familiar with quantifiers and predicates
Learn various proof techniques
Explore what an algorithm is
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4. Mathematical Logic
Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid
Theorem: a statement that can be shown to be true (under certain conditions)
Example: If x is an even integer, then x + 1 is an odd integer
This statement is true under the condition that x is an integer is true
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5. Mathematical Logic
A statement, or a proposition, is a declarative sentence that is either true or false, but not both
Lowercase letters denote propositions
Examples:
p: 2 is an even number (true)
q: 3 is an odd number (true)
r: A is a consonant (false)
The following are not propositions:
p: My cat is beautiful
q: Are you in charge?
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6. Mathematical Logic Truth value Negation
One of the values “truth” or “falsity” assigned to a statement
True is abbreviated to T or 1
False is abbreviated to F or 0
Negation
The negation of p, written ∼p, is the statement obtained by negating statement p
Truth values of p and ∼p are opposite
Symbol ~ is called “not” ~p is read as as “not p”
Example:
p: A is a consonant
~p: it is the case that A is not a consonant
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7. Mathematical Logic Truth Table Conjunction
Let p and q be statements.The conjunction of p and q, written p ^ q , is the statement formed by joining statements p and q using the word “and”
The statement p∧q is true if both p and q are true; otherwise p∧q is false
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8. Mathematical Logic
Conjunction
Truth Table for
Conjunction:
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9. Mathematical Logic Disjunction
Let p and q be statements. The disjunction of p and q, written p ∨ q , is the statement formed by joining statements p and q using the word “or”
The statement p∨q is true if at least one of the statements p and q is true; otherwise p∨q is false
The symbol ∨ is read “or”
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10. Mathematical Logic
Disjunction
Truth Table for Disjunction:
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11. Mathematical Logic Implication
Let p and q be statements.The statement “if p then q” is called an implication or condition.
The implication “if p then q” is written p  q
p  q is read:
“If p, then q”
“p is sufficient for q”
q if p
q whenever p
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12. Mathematical Logic Implication Truth Table for Implication:
p is called the hypothesis, q is called the conclusion
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13. Mathematical Logic Implication
Let p: Today is Sunday and q: I will wash the car. The conjunction p  q is the statement:
p  q : If today is Sunday, then I will wash the car
The converse of this implication is written q  p
If I wash the car, then today is Sunday
The inverse of this implication is ~p  ~q
If today is not Sunday, then I will not wash the car
The contrapositive of this implication is ~q  ~p
If I do not wash the car, then today is not Sunday
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14. Mathematical Logic Biimplication
Let p and q be statements. The statement “p if and only if q” is called the biimplication or biconditional of p and q
The biconditional “p if and only if q” is written p  q
p  q is read:
“p if and only if q”
“p is necessary and sufficient for q”
“q if and only if p”
“q when and only when p”
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15. Mathematical Logic Biconditional Truth Table for the Biconditional:
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16. Mathematical Logic Statement Formulas Definitions
Symbols p ,q ,r ,...,called statement variables
Symbols ~, ∧, ∨, →,and ↔ are called logical connectives
A statement variable is a statement formula
If A and B are statement formulas, then the expressions (~A ), (A ∧ B) , (A ∨ B ), (A → B ) and (A ↔ B ) are statement formulas
Expressions are statement formulas that are constructed only by using 1) and 2) above
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17. Mathematical Logic Precedence of logical connectives is:
~ highest
∧ second highest
∨ third highest
→ fourth highest
↔ fifth highest
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18. Mathematical Logic Example:
Let A be the statement formula (~(p ∨q )) → (q ∧p )
Truth Table for A is:
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19. Mathematical Logic Tautology Contradiction
A statement formula A is said to be a tautology if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A
Contradiction
A statement formula A is said to be a contradiction if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A
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20. Mathematical Logic Logically Implies Logically Equivalent
A statement formula A is said to logically imply a statement formula B if the statement formula A → B is a tautology. If A logically implies B, then symbolically we write A → B
Logically Equivalent
A statement formula A is said to be logically equivalent to a statement formula B if the statement formula A ↔ B is a tautology. If A is logically equivalent to B , then symbolically we write A ≡ B (or A ⇔ B)
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21. Mathematical Logic
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22. Next slide, adapted from National Taiwan University
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