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Level 376

Discrete Functions


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Sequence
A set of numbers arranged in a special order or pattern.
Slope is the first difference (m)
How do you find create a formula using first differences (linear)?
Half of the second difference is the a-value
How do you create a formula with second differences (quadratic)?
Fibonacci sequence
Each number, after the first two numbers is the sum of the previous two numbers
Arithmetic sequences
The common difference d, between each consecutive term is constant; each term is formed by adding a fixed quantity to the term before it
Recursion formula
A formula in which each term of a sequence is generated from the preceding term or terms
tn=tn-1 - x
What is the general recursion formula?
Pascals triangle
An arrangement of numbers. Each row is generated by calculating the sum of pairs of consecutive terms in the previous row. Row 0 has only a 1.
tn,m where
Positions of terms in pascals triangle
tn-1,m-1 , tn-1,m
What is the sum of tn,m?
Common difference (d)
Corresponds to the difference/rate of change between two consecutive terms in an arithmetic sequence
Arithmetic series
Sum of the terms of a sequence
Partition
Let A be a set. A partition of (or on) A is a set of nonempty, pairwise disjoint (no element is common between two parts) sets whose union is A.
Binomial coefficient
Let n, k be integers. (n / k) denotes the number of k-element subsets of an n-element set. "n choose k (things)" (n / k) = n!/k!(n-k)!
Pascal's Identity
(n / k) = (n-1 / k-1) + (n-1 / k)
Proof by Contrapositive
A direct proof of an implication's contrapositive. This uses the fact that for every two statements P and Q, the implication P → Q and its contrapositive are logically equivalent.
Proof by contradiction
To prove "if A, then B," assume A and not B - look for a contradiction that then proves B. (Good for proving an empty set or uniqueness)
function
If you repeat an INPUT, and you always get the same OUTPUT, your data is a ______________________ .
Domain, Image
The domain is the set of all possible first elements of the ordered pairs in f - dom f. The image is the set of the second elements - im f.
One-to-one
A function f is called one-to-one provided that, whenever (x, b), (y, b) ∈ f, we must have x=y. In other words, if x≠y, then f(x)≠f(y). (Has to pass the horizontal line test). Inverse func…
Onto
Let f: A → B. f is onto B provided that for every b in B, there is an a in A so that f(a)=b. in other words, im f=B.
Bijection
The function is both one-to-one and onto.
(a,a) for every a in the set
Ex. Sn = t1 + t2 + ... + tn-1 + tnReflexive
Symmetric
Flip the left side of the equation to the right side.
Antisymmetric
(a,b) and (b,a) ONLY if a=b
Asymetric
(a,b) implies NOT(b,a)
Transitive
(a,b) and (b,c) implies (a,c)
Reflexive
(a,a) for every a in the set