**SLR 511F: 10 MATHEMATICAL IDEAS EVERYONE SHOULD KNOW and their everyday applications in the real world **

** John Clements.**

** SESSION # 4, Mon. Oct. 3/11. **

** SESSION #4 will begin with a Review of the Numbers topic
including a very brief look at the History of Numbers, take a look at Diana Pugdley's
interesting application of mathematics to rowing (a boat)
and have a look at TOPIC #7 The Concept of PROOF. **

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** The Concept of PROOF is not only the foundation of mathematics but it is how
we convince ourselves that something is true in everyday decision making,
e.g. making predictions, understanding legal arguments, making medical diagnoses,etc.**

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** TOPIC # 7: The concept of PROOF **

** Given a mathematical statement (Theorem, etc.), there are 4 types of MATHEMATICS PROOFS to establish its validity or its falseness: **

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** (i) The Direct Method. Follow a sequence of logically valid statements
to the conclusion (i.e. a => b => c etc. Theorem 1, Theorem 2 **

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** (ii) The Indirect Method. Assume the result is true and then deduce a
contradiction - then it is false. (i.e. proof by contradiction). Theorem 3 **

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** (iii) Method of Induction. Show that a sequence of Propositions Pn is true
for n=1 and then establish that that implies it must be true for
every n=2,3,4,....(i.e. P1 => P2 => P3 etc.) Theorem 4 **

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** (iv) By Counterexample. Find a result that refutes the statement. Theorem 5 **

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** Each of these approaches will be examined.**

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