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# lecture12 - Lecture 12 Functions Sequences and Summations...

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Lecture 12 Functions; Sequences and Summations

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Recap Functions: Mapping from the domain to co-domain Every element in the domain must be mapped to a unique element one-to-one, onto Bijection Inverse Composition
Ceiling and floor properties Let n be an integer (1a) x = n if and only if n ≤ x < n+1 (1b) x = n if and only if n-1 < x ≤ n (1c) x = n if and only if x-1 < n ≤ x (1d) x = n if and only if x ≤ n < x+1 (2) x-1 < x ≤ x ≤ x < x+1 (3a) -x = - x (3b) -x = - x (4a) x+n = x +n (4b) x+n = x +n

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Ceiling property proof Prove rule 4a: x+n = x +n Where n is an integer Will use rule 1a: x = n if and only if n ≤ x < n+1 ( in both directions ) Direct proof: Let m = x Thus, m ≤ x < m+1 (by rule 1a) Add n to both sides: m+n ≤ x+n < m+n+1 By rule 1a, m+n = x+n Since m = x , m+n also equals x +n Thus, x +n = m+n = x+n
Factorial is denoted by n! n! = n * (n-1) * (n-2) * … * 2 * 1

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lecture12 - Lecture 12 Functions Sequences and Summations...

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