1. 1 Math 306 Foundations of Mathematics I Math 306 Foundations of Mathematics I Goals of this class Introduction to important mathematical concepts Development of mathematical reasoning skills Study of formal proof techniques Discussion of applications

2. 2 Outline of Topics Mathematical Logic Proof Techniques Mathematical Induction Set Theory Functions Relations

3. 3 Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.

4. 4 Example Sequence of statements: 1)All students take Math306. 2)Anyone who takes Math306 is a Math major. 3)Therefore, all students are Math majors. If (1) and (2) were true, then logic would assure that (3) is true.

5. 5 Outline of logic topics Simple Statements Compound Statements Conditional Statements Quantified Statements Valid and Invalid Arguments for all kind of statements

6. 6 Logical Statements Definition: A statement is a sentence that is true or false but not both. Examples: 3+5=8 (true statement) Today is Friday (false statement) Note: x>y is not a statement

7. 7 Logical Connectives For given statements p and q: Negation of p: ~p (not p) Conjunction of p and q: ( p and q) Disjunction of p and q: (p or q)

8. 8 Truth table for negation p~p TF FT

9. 9 Truth table for conjunction pq TTT TFF FTF FFF

10. 10 Truth table for disjunction pq TTT TFT FTT FFF

11. 11 Statement form Expression made up of statement variables (such as p,q) and logical connectives; becomes a statement when actual statements are substituted for the variables. Example: (Exclusive Or)

12. 12 Truth Table for a Statement Form Ex: Truth table for pq~p TTFTF TFFTF FTTTT FFTFF

13. 13 Logical equivalence Statements P and Q are logically equivalent: if and only if they have identical truth values for each substitution of their component statement variables. Ex:

14. 14 Verifying logical equivalence Ex: pq~p~q TTFFTFF TFFTTFF FTTFTFF FFTTFTT

15. 15 Important Logical Equivalences Double negation: De Morgan’s laws: Ex: negation of -5 < x < 7 is

16. 16 Tautologies and Contradictions Tautology is a statement form which is true for all values of statement variables. E.g., is a tautology: Contradiction is a statement form which is false for all values of statement variables. E.g., is a contradiction:

17. 17 More Logical Equivalences Commutative laws: Associative laws: Distributive laws: Absorption laws:

18. 18 Simplifying Statement Forms