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Class Notes for Mathematics at University of Toronto Mississauga (UTM)

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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 1: Quadratic Formula

181
Mat102: mathematical thinking-lecture 1: ch. 1 numbers,sets and functions. Theorem( the quadratic formula): ax^2+bx+c=0 (for all a 0) has 3 situations:
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 21: Modular Arithmetic, Equivalence Class, If And Only If

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[5]={,-7,-3,1,5,9,13,} on z: ab iff a-b is divisible by 4. Theorem: let ~ be an equivalence relation on a set s. then every element belongs to some equ
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture 19: GCD

116
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 15: Recursive Definition, Mathematical Induction

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L-tiling (i. e can be covered by l-shapes ). a checkerboard with one square removed has an. Proof : base case: for n=1, we have 2 2 board with 1 square
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 20: Modular Arithmetic, New Zealand, If And Only If

120
Often, we need to compare two objects in a given set, and decide whether they do or do not satisfy a certain property ( or condition). If a and b are s
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 2: If And Only If

103
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture 5: Sets Part 2

64
If a, b r with a b, we define: [a,b]={x r a x b} (closed interval) (a,b)={x r a< x<b} (open interval) Set operations: for sets a and b, we define: Inte
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 4: Empty Set

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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 6: If And Only If, Bounded Function, Farad

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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture 18: Cardinality (cont'd)

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Definition: a set is called countable if it has the same cardinality as n. Example: n, 2n, z and q are countable. (n n is countable--theorem 4. 44 page
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 17: Rational Number, Surjective Function, Bijection

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The composition of g with f, denoted g o f, is the function from a to c, given by (g o f)(x)=g(f(x)) for x a. )=e f o g:r r,f o g(x)=f(g(x))=f(e )= f o
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UTMMAT102H5Alexander RennetSummer

MAT102H5 Lecture Notes - Lecture 16: Surjective Function, Bijection, Injective Function

79
Example: consider the following function f:a b, given by. Answer: no, for two reasons: the element g has two images, which is not allowed, the elements
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