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Mathematical shorthand: the symbols that are just words

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Learning how to learn

Mathematical shorthand: the symbols that are just words

Math looks dense because it packs ordinary sentences into single symbols. x ∈ S is three characters for “x is an element of S”—nothing more. Every symbol in these articles stands for a word or phrase you already know, only written shorter. This page decodes the shorthand used across the site when a symbol slows you down.

Do not memorize the table. Use the trick: when a formula stops you, translate each symbol into its word and continue. The logic underneath matters. The symbols are only the font.

The one move: read a symbol as its word

You already do this with arithmetic. + is not a glyph to puzzle over—you read it as “plus” and continue. The symbols below work the same way; they are simply less familiar. When you see a ∈ B, read it aloud as “a is in B” and continue the sentence. Translate each symbol, and the formula becomes prose.

Sets: the curly-bracket shorthand

A set is a collection of things. Its notation is where many readers first stumble. Each piece is one word:

Symbol Say it Code spelling
{1, 2, 3} “the set of 1, 2, and 3” a list of values
x ∈ S “x is in S” S.includes(x)
x ∉ S “x is not in S” !S.includes(x)
A ∪ B “A or B” (everything in either) the union of two sets
A ∩ B “A and B” (everything in both) the intersection
∅ “the empty set” [] / {}
B ⊂ A “B is a subset of A” every element of B is in A

One piece of set notation deserves a closer look. It is a list comprehension in disguise. Set-builder notation defines a set with a rule:

A = {x | x < 20 and x is even}

Read this as “A is the set of all x such that x is less than 20 and x is even”—the same idea as for x <- 1..20, rem(x, 2) == 0, do: x or array.filter(x => x < 20 && x % 2 === 0). The bar | means “such that”.

Functions: the arrow shorthand

Functions use two pieces of notation. Read them aloud and both become simple.

f(x) reads “f of x”: apply the function to input x. f : S → T reads “f maps S to T”—S is the set of inputs (the domain), T is the set of outputs (the codomain), and the arrow means “maps to”. f(s) = t reads “f sends s to t”. The code spelling of f : S → T is a type signature like (x: S) => T.

Composition uses the circled symbol: g ∘ f reads “g after f”, meaning g(f(x)). The little circle ∘ means “compose”. Read it right-to-left: f runs first, then g runs on f’s result. In code, g ∘ f is x => g(f(x)).

The arrow that means three things

The arrow → is the most overloaded symbol on this site. Let the context decide what it means. It has three uses:

  • “maps to” in functions - f : S → T (f maps S to T).
  • “implies” in logic - a → b (if a, then b).
  • “becomes” in rewriting - 2 * x + 1 → 2 * 3 + 1 → 7 (each step reduces to the next).

The same arrow has three words. The surrounding sentence tells you which word applies. That is the rule for shorthand: a symbol has no single fixed meaning; context pins it down.

Logic: two fonts for the same five words

This is the family where the site uses the friendlier form. The logic exercises and the truth-tables article write the connectives in the code you already type - &&, ||, !, ->, <-> - rather than the math symbols ∧, ∨, ¬, →, ↔. They express the same five ideas in a different font:

Word Code spelling (this site) Math symbol
and a && b a ∧ b
or a || b a ∨ b
not !a ¬a
implies a -> b a → b
if and only if a <-> b a ↔ b

When another text uses the math-font versions, swap them for the code versions: ∧ is &&, ∨ is ||, and so on. Two more symbols appear in the proofs and induction material: ∀ reads “for all”, and ∃ reads “there exists”. ∀x, P(x) means “for every x, P holds”; ∃x, P(x) means “there is some x where P holds”. On this site, those usually appear as the words “for all” and “there exists” instead. The claim is the same.

Lambda: the last symbol on the site

λx. M reads “the function of x that returns M”. The Greek letter λ (lambda) means “function of”, and the dot separates the input from the body. So λx. x is “the function of x that returns x”—the identity function—and λx. λy. x is “the function of x that returns the function of y that returns x”. It is an arrow function written with a Greek letter: λx. M is x => M.

You already know how to read this

No symbol here names an idea you have not seen. ∈ is “in”, ∪ is “or”, ∘ is “after”, → is “maps to” or “implies” or “becomes”, and λ is “function of”. The symbol is not the main difficulty. The concept is, and you usually met that concept in code. So use this strategy: do not stop at a symbol. Translate it into its word and keep moving. If a symbol is genuinely new, such as the first time you meet λ, treat it as one more word. The surrounding article will tell you what it means.

Where to go next

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Exercises