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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

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

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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.

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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.

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

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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

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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)

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8 Truth table for negation p~p TF FT

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9 Truth table for conjunction pq TTT TFF FTF FFF

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10 Truth table for disjunction pq TTT TFT FTT FFF

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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)

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

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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:

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

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

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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:

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

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18 Simplifying Statement Forms

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