2. Mathematical language for group theory

Read phrases such as “Let A be a set” and symbols such as := as instructions that create names for objects. Practice translating short definition lines into plain English before using the notation that follows.

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Use what you learned in the previous lesson to solve real-world problems.

Work with sets as collections of objects and elements as the objects inside them. Practice reading and writing membership statements with ∈ and ∉, such as a ∈ A or 3 ∉ {1, 2}.

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Use what you learned in the previous lesson to solve real-world problems.

Use roster notation like {1, 2, 3} and set-builder notation like {x ∈ ℤ : x is even}. Practice reading the colon or vertical bar as “such that” and choosing the clearer notation for a small example.

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Use what you learned in the previous lesson to solve real-world problems.

Read A ⊆ B as “every element of A is also in B,” and distinguish subset from set equality. Practice checking A = B by comparing membership in both directions.

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Use what you learned in the previous lesson to solve real-world problems.

Distinguish the empty set ∅ from a singleton set like {a}. Practice spotting common traps such as ∅ versus {∅} and a versus {a}.

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Use A ∪ B, A ∩ B, A \ B, and complements to describe how sets combine or differ. Practice translating each symbol into a simple membership rule about which elements are included.

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Recognize the standard symbols ℕ, ℤ, ℚ, ℝ, and ℂ and the kinds of numbers they name. Practice placing example numbers into the smallest familiar number set that contains them.

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Read ∀ as “for all” and ∃ as “there exists,” including the useful symbol ∃! for “there exists exactly one.” Practice translating quantified sentences into plain English and identifying the variable being quantified.

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Read and, or, not, ⇒, and ⇔ in short mathematical statements. Practice telling the difference between “if P then Q,” “P if and only if Q,” and the negation of a simple claim.

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Treat (a, b) as an ordered pair where order matters, and extend the same idea to tuples like (a, b, c). Practice reading A × B as the set of all ordered pairs with first entry from A and second entry from B.

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Use what you learned in the previous lesson to solve real-world problems.

Use subscripts such as a_i and ellipses such as a_1, a_2, …, a_n to describe patterned lists without writing every item. Practice identifying what the index is allowed to be and what object each indexed symbol names.

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Use what you learned in the previous lesson to solve real-world problems.

Read f: A → B as a function with domain A and codomain B, and read f(a) as the value of f at input a. Practice separating the function itself, its input, and its output.

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Recognize that a function must assign exactly one output to each input in its domain. Practice using a small table or graph of ordered pairs to decide whether a proposed rule is well-defined as a function.

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Read f(S) as the image of a subset S and f⁻¹(T) as the preimage of a subset T, even when f has no inverse function. Practice tracing which outputs are hit and which inputs land inside a chosen target set.

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Compare injective, surjective, and bijective functions using finite examples. Practice recognizing one-to-one behavior, onto behavior, and the case where every codomain element is hit exactly once.

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Read g ∘ f as “do f first, then g,” and recognize the identity function id_A as the function that leaves every element of A unchanged. Practice computing a simple composition and checking that the domains and codomains match.

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Read a map like ⋆: A × A → A as a rule that takes two inputs from A and returns one output in A. Practice translating between function notation ⋆(a, b) and infix notation a ⋆ b without using group axioms yet.

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Use what you learned in the previous lesson to solve real-world problems.

Review this chapter with practice based on your mistakes.