L2 - MATH 135 Lecture II Notes Fall 2008 Patterns and...

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MATH 135 Fall 2008 Lecture II Notes Patterns and Conjectures We often use patterns in mathematics to guess what is happening in general (that is, make a “con- jecture”), and then try to prove our pattern/conjecture (that is, turn it into a theorem). Consider the following problem: n points are chosen on the circumference of a circle in such a way that when all pairs of points are connected, no three of these lines intersect at a single point. How many regions are formed inside the circle? Types of Statements to Prove There are two basic types of statements that we try to prove in mathematics: 1) Simple statements: A eg. “ 2 is irrational” 2) Conditional statements: A B (“ A implies B ” or “If A then B ”) eg. “If n is an odd integer, then n 2 - 1 is divisible by 4.” Here, A = “ n is an odd integer” is our hypothesis and B = “ n 2 - 1 is divisible by 4” is our conclusion. A and B are both statements on their own about some unknown number n . To prove propositions of Type 1, we use the axioms of mathematics, logic, our ingenuity, and other results we know to be true. To prove propositions of Type 2, we use the hypothesis (or hypotheses) and the list of tools above. In doing so, we assume that
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L2 - MATH 135 Lecture II Notes Fall 2008 Patterns and...

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