#math side blog

[without-loss-of-generality]

definition: divergent

a sequence is said to be divergent if it does not converge. that is, there is no single number which the sequence is tending towards. example: {Xn} = 1, 2, 3, 4, 5, 6, 7, ... and so on does not converge, it just keeps getting larger and larger

[without-loss-of-generality]

definition: sequence

a sequence is really just a list of numbers, where the next number in the list is often determined by some sort of formula. we ususally denote an entire sequence by {Xn}, and the nth by Xn. for example, the 5th number in the sequence {Xn} would be X5.

[without-loss-of-generality]

definition: injection

an injection is a function f from a set X to a set Y with the special property that for two points, x ∈ X and y ∈ Y, x = y if and only if f(x) = f(y)

take for example f(x) = x + 2. take x = 3. we see that f(3) = 5. there is no other value of x that makes f(x) equal to 5, so if we have that f(a) = 5 and f(b) = 5, we know that a = b = 3.

now take g(x) = x². take x = -1. g(-1) = 1, but g(1) also equals 1. hence g(x) = x² is not an injection.

an injection can also be called a one-to-one function, or that it passes the horizontal line test.

[without-loss-of-generality]

definition: function

formally, a function from a set X to a set Y is a subset of the Cartesian product of X and Y. so, a function is a set of points (x, y). also, a function must have the property that for every element x’ ∈ X, there is only one y’ ∈ Y such that the point (x’, y’) is in f. you probably talked about this property in high school as passing the ‘vertical line test.’