MATHEMATICAL REASONING

MATHEMATICAL REASONING

1. Methods

   • Direct Proof
     • Outline
       • Proposition: If P, Then Q
         • Example
     • Strategy
       • The path from assumption (p) to the conclusion (q) is obvious
       • Assumed axioms and previously proved theorem
   • Mathematical Induction
     • Outline
       • Proposition: The Stattement S1, S2, S3,,, are all true
         • Example
     • Strategy
       • When the universe of discourse for a predicate quantified by "for every" is the set of all positive integers
   • Indirect Method
     • Contrapositive
       • Outline
         • Preposition: If P, Then Q
           • Example
       • Truth Table
         • • • If the original statement is TRUE, the contrapositive is TRUE. If the original statement is FALSE, the contrapositive is FALSE
       • Strategy
         • The conclusion(Q) of proposition has simpler logical form than the hypothesis(P).
         • Assumption that P does not match with the "If part"
         • Hint:Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.
     • Contradiction
       • Outline
         • Proposition: P
           • Example
       • Truth Table
       • Strategy
         • Any proof of a theorem in set theory in which the conclusion asserts that some set equals the empty set

2. Statement

   • Usually in Form of
     • Biconditional Statement
       • P if and only if Q
     • Conditional Statement
       • If P, then Q
   • Type of Statement
     • Known to be True
     • Truth Unknown
     • Know to be False

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