Basic Set Symbols
| Symbol | Name | Meaning | Example |
|---|---|---|---|
{ } | Set braces | Defines a set | {1, 2, 3} |
∈ | Element of | Is an element of | 2 ∈ {1,2,3} |
∉ | Not element of | Is not an element of | 4 ∉ {1,2,3} |
= | Equality | Two sets are equal | {1,2} = {2,1} |
≠ | Not equal | Sets are not equal | {1} ≠ {1,2} |
Subsets & Supersets
| Symbol | Name | Meaning | Example |
|---|---|---|---|
⊆ | Subset (or equal) | Every element of A is in B | {1} ⊆ {1,2} |
⊂ | Proper subset | Subset but not equal | {1} ⊂ {1,2} |
⊇ | Superset | Contains another set | {1,2} ⊇ {1} |
⊃ | Proper superset | Superset but not equal | {1,2} ⊃ {1} |
Set Operations
| Symbol | Name | Meaning | Example |
|---|---|---|---|
∪ | Union | Elements in A or B | {1,2} ∪ {2,3} = {1,2,3} |
∩ | Intersection | Elements in both A and B | {1,2} ∩ {2,3} = {2} |
\ | Set difference | Elements in A not in B | {1,2,3} \ {2} = {1,3} |
Δ | Symmetric difference | In A or B but not both | {1,2} Δ {2,3} = {1,3} |
∅ | Empty set | Set with no elements | ∅ ⊆ A |
Universal & Complement
| Symbol | Name | Meaning | Example |
|---|---|---|---|
U | Universal set | All possible elements | A ⊆ U |
Aᶜ or ¬A | Complement | Elements not in A | Aᶜ = U \ A |
Set Size & Construction
| Symbol | Name | Meaning | Example |
|---|---|---|---|
|A| | Cardinality | Number of elements in A | {1,2,3} ⇒ A = 3 |
{ x | P(x) } | Set-builder notation | All x satisfying property P(x) | { x | x > 0 } = {1,2,3,...} |
℘(A) | Power set | The set of all subsets of A | ℘({1}) = {∅, {1}} |
Logic Symbols (Used with Sets)
| Symbol | Name | Meaning | Example |
|---|---|---|---|
∀ | For all | Universal quantifier | ∀x ∈ A |
∃ | There exists | Existential quantifier | ∃x ∈ A |
∧ | And | Logical AND | x ∈ A ∧ x ∈ B |
∨ | Or | Logical OR | x ∈ A ∨ x ∈ B |
¬ | Not | Logical negation | ¬(x ∈ A) |
⇒ | Implies | Logical implication | x ∈ A ⇒ x ∈ B |
⇔ | If and only if | Logical equivalence | A ⊆ B ⇔ ∀x (x∈A ⇒ x∈B) |
Common Number Sets
| Symbol | Name |
|---|---|
ℕ | Natural numbers |
ℤ | Integers |
ℚ | Rational numbers |
ℝ | Real numbers |
ℂ | Complex numbers |
Core Set Theory & Logic Symbols
| Symbol | Name | Plain-English Meaning | Example |
|---|---|---|---|
∈ | Element of | An object belongs to a set | 3 ∈ {1,2,3} |
∉ | Not element of | An object does not belong to a set | 4 ∉ {1,2,3} |
⊆ | Subset (or equal) | Every element of A is also in B | {1,2} ⊆ {1,2,3} |
∪ | Union | Elements in A or B (or both) | {1,2} ∪ {2,3} = {1,2,3} |
∩ | Intersection | Elements in both A and B | {1,2} ∩ {2,3} = {2} |
\ | Set difference | Elements in A but not in B | {1,2,3} \ {2} = {1,3} |
∅ | Empty set | Set with no elements | ∅ ⊆ A |
|A| | Cardinality | Number of elements in A | |{1,2,3}| = 3 |
∀ | For all | Statement applies to every element | ∀x ∈ A, x > 0 |
∃ | There exists | At least one element satisfies | ∃x ∈ A, x = 0 |
⇒ | Implies | If left is true, right must be true | x ∈ A ⇒ x ∈ B |