Mathematical Notation Lookup Table

After struggling with mathematical notation, I found that it helps to think of it as a mathematician's shorthand: a concise way to express the relationships between variables. It's also similar to a programming language: a format to communicate steps for computation. Here, I've listed some mathematical notation beyond what I learned in high school. I grouped them according to whatever umbrella term that I can find and their applications. Unlike most of the tables I found online, I try to explain each piece of notation in non-formal terms. This list is far from comprehensive, but it will expand as I encounter new mathematical notation.
Disclaimer: I am not a mathematician, not even a math major. So, this table may contain some misconceptions and I would be happy if anyone corrects me on them.


Accumulation Notation

(a name I made up)
Applications: Series, Probability
Representation Pronunciation Interpretation
Σx=nm f(x) Sigma This is a summation, whic means means inputting all the integers from n to m into the function f, then taking the sum of the outputs
Πx=nm f(x) Pi input all the integers from n to m into the function f, then take the product of the outputs


Calculus Notation

Applications: Physics, Chemistry
Representation Pronunciation Interpretation
δx differential of x an infinitely small increment of a variable x. On a continuous line, the point right next to x = 1 is x = 1+δx
Δx delta x unlike the differential, this represents a finite increment of x as Δx = (x2-x1)
δxδy derivative of x with respect to y This is how much x changes in one instant as y changes. On the function in the usual x and y graph, this would represent the exact slope at one specific point.
f(x) δx Integral of f(x) An integral is like a summation for continuous functions. In this case, the integral equals the area beneath the line/curve. In other cases, it can also mean the length of the line/curve, or volume.
Contour Integral This is an integral over a closed loop


Set Notation

Applications: Probability, Statistics
Representation Pronunciation Interpretation
X = {a,b,c} set a,b, and c are elements of set X
all real numbers include all rational and irrational numbers
all integers both positive and negative whole numbers
all natural numbers all positive integers, and sometimes 0
fractions ratio of two integers
{p|q} "such that"
"given"
the set of all p that follows the rules defined by q
x ∈ ℤ "is within" x is within a set of all integers
a ⊂ b "subset" a is a subset of b
b ⊄ a "non-subset" b is not a subset of a
C ∪ D "union" a set with all the elements from sets C and D, but with no repeating values
C ∩ D "intersection" a set with only elements in both sets C and D

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