Symbol Quick Reference
Every symbol used in this course. The Say it as column shows exactly how to pronounce each symbol when reading mathematics aloud or writing Canvas discussions.
Logic & Proofs — Modules 2–5
| Symbol | Say it as | Meaning | Example | |
|---|---|---|---|---|
| ∧ | and | Logical conjunction | p ∧ q | |
| ∨ | or | Logical disjunction | p ∨ q | |
| ¬ | not | Logical negation | ¬p | |
| → | implies / if…then | Conditional | p → q | |
| ↔ | if and only if | Biconditional | p ↔ q | |
| ≡ | is equivalent to | Logical equivalence | p ≡ q | |
| ∀ | for all / for every | Universal quantifier | ∀x ∈ ℤ | |
| ∃ | there exists | Existential quantifier | ∃x ∈ ℤ | |
| ∴ | therefore | Conclusion marker | ∴ p is true |
Set Theory — Modules 1, 10
| Symbol | Say it as | Meaning | Example | |
|---|---|---|---|---|
| ∈ | is an element of / in | Membership | 3 ∈ {1,2,3} | |
| ∉ | is not an element of | Non-membership | 4 ∉ {1,2,3} | |
| ⊆ | is a subset of | Subset (may be equal) | {1,2} ⊆ {1,2,3} | |
| ⊊ | is a proper subset of | Subset and not equal | {1,2} ⊊ {1,2,3} | |
| ∪ | union | All elements in A or B | A ∪ B | |
| ∩ | intersect | Elements in both A and B | A ∩ B | |
| ∅ | empty set | Set with no elements | A ∩ Aᶜ = ∅ | |
| ℤ ℝ ℕ ℚ | integers / reals / naturals / rationals | Standard number sets | x ∈ ℤ |
Functions, Relations & Counting — Modules 11–14
| Symbol | Say it as | Meaning | Example | |
|---|---|---|---|---|
| f: A→B | f maps A to B | Function declaration | f: ℤ→ℤ | |
| ∘ | composed with | Function composition | f ∘ g | |
| C(n,r) | n choose r | Combinations | C(10,3)=120 | |
| P(n,r) | P of n comma r | Permutations | P(10,2)=90 | |
| Σ | sum / sigma | Summation notation | Σᵢ₌₁ⁿ i | |
| ≡ (mod n) | is congruent to … mod n | Modular congruence | 17 ≡ 2 (mod 5) |
How to Type Symbols
Use the copy buttons above when possible. When you need to type from scratch:
Windows
- Windows key + . → opens emoji & symbol picker — search "element", "union", etc.
- Character Map app → search for any symbol by name
- Word/Canvas equation editor → use LaTeX shortcodes (see Section 6)
Mac
- Control + Command + Space → opens Special Characters viewer
- Edit menu → Emoji & Symbols → same viewer from any app
- Search "not equal", "intersection", "for all" etc.
📋 Copy-paste cheat sheet — bookmark this page
Glossary by Module
M1 & M10 Set Theory
Set
A well-defined collection of distinct objects called elements. Written with curly braces: {1, 2, 3}.
Element / Member
An object that belongs to a set. We write x ∈ A to say x is an element of A.
Subset (⊆)
A is a subset of B if every element of A is also in B. A set is always a subset of itself.
Proper Subset (⊊)
A is a proper subset of B if A ⊆ B and A ≠ B. A proper subset is strictly smaller.
Cardinality
The number of elements in a set. Written |A|. Duplicates count once: |{1,1,2}| = 2.
Cartesian Product (A × B)
The set of all ordered pairs (a, b) where a ∈ A and b ∈ B. |A × B| = |A| · |B|.
Union (∪)
A ∪ B = all elements in A or B or both. Think OR.
Intersection (∩)
A ∩ B = only elements in both A and B. Think AND.
Complement (Aᶜ)
Everything in the universal set U that is NOT in A. Always specify U first.
Power Set
The set of all subsets of A, written 𝒫(A). If |A| = n then |𝒫(A)| = 2ⁿ.
M2–M5 Logic & Proofs
Proposition / Statement
A sentence that is either true or false, never both. Variables p, q, r denote statements.
Tautology
A compound statement that is always true, regardless of truth values of its parts.
Contradiction
A compound statement that is always false.
Contrapositive
The contrapositive of p → q is ¬q → ¬p. Always logically equivalent to the original.
Converse
The converse of p → q is q → p. NOT equivalent to the original in general.
Valid Argument
An argument where if all premises are true, the conclusion must be true. Truth of premises is not required.
Direct Proof
Assume hypothesis true, use definitions and logic to derive the conclusion.
Proof by Contradiction
Assume the negation of what you want to prove, then derive a contradiction.
Counterexample
A single example that makes a universal statement false. One is enough to disprove.
Mathematical Induction
Two steps: Base Case (prove P(n₀)) and Inductive Step (P(k) → P(k+1)). Together they prove P(n) for all n ≥ n₀.
M9 Sequences
Sequence
An ordered list of numbers. Each position has an index: a₁, a₂, a₃, …
Recurrence Relation
A rule that defines each term using previous terms, plus initial conditions.
Arithmetic Sequence
Constant difference d between terms. aₙ = a₁ + (n−1)d. Grows linearly.
Geometric Sequence
Constant ratio r between terms. aₙ = a₁ · rⁿ⁻¹. Grows exponentially.
Explicit Formula
A formula giving aₙ directly from n, without needing previous terms.
Inductive Hypothesis
In an induction proof, the assumption that P(k) holds. Used to prove P(k+1).
M11–M14 Functions, Relations & Counting
Function
Assigns each element of the domain exactly one element in the codomain. Every input has exactly one output.
One-to-One (Injective)
Different inputs always give different outputs. Prove: assume f(a) = f(b), show a = b.
Onto (Surjective)
Every element of the codomain is hit by at least one input. Range = Codomain.
Bijection
One-to-one AND onto. A perfect pairing between domain and codomain.
Equivalence Relation
A relation that is reflexive, symmetric, and transitive. All three must hold.
Countably Infinite
A set that can be put in bijection with the natural numbers — it can be listed completely.
Permutation P(n,r)
Ordered arrangements of r from n. Use when order matters. P(n,r) = n!/(n−r)!
Combination C(n,r)
Unordered selections of r from n. Use when order does not matter. C(n,r) = n!/(r!(n−r)!)
Pigeonhole Principle
If n+1 objects go into n boxes, at least one box has 2 or more. Proves existence without naming the case.
Binomial Theorem
(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ. The coefficients are C(n,k), readable from Pascal's triangle.
Common Mistakes — Watch Out
❌ Wrong
{2} ∈ {1, 2, 3}
The set {2} is not an element of {1,2,3}. The elements are the numbers 1, 2, 3.
✅ Correct
2 ∈ {1, 2, 3}
The number 2 is an element. Write {2} ∈ {1, {2}, 3} only if {2} itself appears.
❌ Wrong
Confusing ⊆ with ∈
∈ is for elements (things). ⊆ is for sets (collections). {2} ⊆ {1,2,3} ✓ and 2 ∈ {1,2,3} ✓ — but not the other way.
✅ Rule
∈ for elements, ⊆ for sets
Ask yourself: "Is what's on the left a thing or a collection?" Thing → ∈. Collection → ⊆.
❌ Wrong
Proving the converse instead of the original
To prove p → q, you must start from p and reach q. Starting from q and reaching p proves the converse, not the theorem.
✅ Rule
State what you assume and what you show
Open every proof step: "Assume p." Close every proof: "Therefore q. □"
❌ Wrong
Skipping the base case in induction
Without the base case, induction proves nothing. The base case anchors the entire chain.
✅ Rule
Always write Step 1 — Base Case explicitly
Show P(n₀) holds by direct substitution before starting the inductive step.
❌ Wrong
Using P when order doesn't matter (or C when it does)
Committee of 3 from 10 → use C(10,3). President and Secretary from 10 → use P(10,2).
✅ Rule
Ask: does the arrangement matter?
Roles, rankings, sequences → P. Groups, committees, hands, subsets → C.
Proof Templates
Use these as checklists when writing Canvas discussion proofs. Every proof you write should follow one of these structures.
Direct Proof — to prove p → q
[Use definitions, algebra, and logic.]
Therefore q. □
Proof by Contrapositive — to prove p → q
Assume ¬q.
[Use definitions and logic to derive ¬p.]
Therefore ¬p.
Since the contrapositive holds, p → q. □
Proof by Contradiction
[Use definitions and logic to derive a contradiction — something known to be false.]
This is a contradiction.
Therefore [original statement] must be true. □
Mathematical Induction
Base Case (n = n₀):
Substitute n₀ into both sides. Show LHS = RHS. ✓
Inductive Step:
Assume P(k) for some k ≥ n₀. [Write the inductive hypothesis explicitly.]
Show P(k+1):
Start with LHS of P(k+1).
Split off the last term.
Apply the inductive hypothesis.
Simplify to match RHS of P(k+1). ✓
By the Principle of Mathematical Induction, P(n) holds for all n ≥ n₀. □
Element Method — to prove A ⊆ B
[Use the definition of A to determine what x satisfies.]
[Show that x must also satisfy the definition of B.]
Therefore x ∈ B.
Since x was arbitrary, every element of A is in B, so A ⊆ B. □
Double Inclusion — to prove A = B
Part 2: Show B ⊆ A. [Use the element method in the other direction.]
Since A ⊆ B and B ⊆ A, we conclude A = B. □
Writing Math in Canvas
Option 1 — Copy and paste symbols
The simplest approach. Use the copy buttons in Section 2 above and paste directly into your Canvas discussion reply.
by definition of ∩, x ∈ A and x ∈ B.
In particular, x ∈ A. □
Option 2 — Canvas Equation Editor
In the Canvas rich-text editor, click the √x button in the toolbar to open the equation editor. Type LaTeX directly:
\cap → ∩ \forall → ∀ \exists → ∃
\neg → ¬ \land → ∧ \lor → ∨
\to → → \equiv → ≡ \emptyset → ∅
If you cannot find a symbol — acceptable plain-text alternatives
Self-Check Checklist
Before submitting any Canvas discussion or assignment, run through the checklist for the relevant module. Check each box mentally.
Set Theory (M1, M10)
- Did I use ∈ for elements and ⊆ for sets (never mixed them)?
- Did I count repeated elements only once in cardinality?
- Did I specify the universal set U before using complement?
- In a set equality proof: did I prove both A ⊆ B and B ⊆ A?
Proofs (M2–M5, M9)
- Did I state clearly what I am assuming and what I am proving?
- In induction: did I write both Base Case and Inductive Step?
- In induction: did I write the inductive hypothesis explicitly?
- Did I end with a conclusion sentence and □?
Counting (M13, M14)
- Did I identify whether order matters (P) or not (C)?
- For multi-group problems: did I multiply independent counts?
- For repetition problems: did I use C(n+r−1, r)?
- Did I use symmetry C(n,r) = C(n,n−r) to simplify?
Functions & Relations (M11, M12)
- For one-to-one: did I assume f(a)=f(b) and show a=b?
- For onto: did I let y be arbitrary in B and construct x?
- For equivalence: did I verify all three properties?
- For modular arithmetic: is the remainder always non-negative?