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# Proofs - CSS342 Proofs Professor Munehiro Fukuda CSS342...

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CSS342: Proofs 1 CSS342: Proofs Professor: Munehiro Fukuda

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CSS342: Proofs 2 Terminologies Axioms: are assumed true. Ex: Given two distinct points, there is exactly one line that contains them. Undefined terms: implicitly defined (and used) by the axioms. Ex: Points, lines Definitions: used to create new concepts. Ex: Two lines are parallel if they never cross each other. Theorem: a proposition that has been proved to be true. If two sides of a triangle are equal, the angles opposite them are equal. Corollary: a theorem that follows quickly from another theorem. Ex: If a triangle is equilateral, then it is equiangular. Lemma: a theorem not interesting but useful in proving another theorem. Ex: A positive integer – 1 ≥ 0
CSS342: Proofs 3 Axioms, Definitions, and Undefined Terms Examples Euclidean geometry: Axiom 1: Given two distinct points, there is exactly one line that contains them. Axiom 2: If three points are not collinear, then there is exactly one plane that contains them. Definition 1: Two angles are supplementary if the sum of their measures is 180. Definition 2: Two lines are parallel if they never cross each other. Undefined terms: points, lines, planes, and angles Real numbers: Axiom 1: The commutative law stands up right for +-*/ operations. Axiom 2: If x and y are in a subset P, -x and –y are not in P, and x+y and xy are in P. Definition 1: P is called positive real numbers. Definition 2: Given a nonnegative real number, x and a positive integer, n, x 1/n is y satisfying y n = x Undefined terms: numbers and 0

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CSS342: Proofs 4 Theorem and Corollary Examples Theorem: If two sides of a triangle are equal, then the angles opposite
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Proofs - CSS342 Proofs Professor Munehiro Fukuda CSS342...

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