Comparison: Logical Operators vs. Set Theory Symbols
April 10, 2025•370 words
Comparison: Logical Operators vs. Set Theory Symbols
1. Functional Purpose
| Category | Purpose |
|---|---|
| Logical Operators | Manipulate truth values of propositions (T/F logic) |
| Set Theory Symbols | Describe relationships between sets and elements |
2. Symbol Comparisons (Similar Meaning, Different Domain)
| Logic Symbol | Set Theory Symbol | Logic Meaning | Set Theory Meaning |
|---|---|---|---|
| ∧ (AND) | ∩ (Intersection) | A and B are both true | Elements common to sets A and B |
| ∨ (OR) | ∪ (Union) | At least one of A or B is true | Elements in A, B, or both |
| ¬A (NOT A) | Aᶜ or A′ (Complement) | A is false | Elements not in set A |
| → (Implication) | — | If A is true, then B must be true | No direct equivalent |
| ↔ (Biconditional) | = | A and B are logically equivalent | Sets A and B have identical elements |
3. Unique to Logic
| Symbol | Name | Meaning |
|---|---|---|
| → | Implication | If A, then B |
| ↔ | Biconditional | A if and only if B |
| ⊤ | Verum | Always true |
| ⊥ | Falsum | Always false |
4. Unique to Set Theory
| Symbol | Name | Meaning |
|---|---|---|
| ∈ | Element of | x ∈ A: x is an element of set A |
| ∉ | Not an element of | x ∉ A: x is not in set A |
| ⊂ | Proper subset | A ⊂ B: All elements of A in B, A ≠ B |
| ⊆ | Subset | A ⊆ B: All elements of A in B |
| ∅ | Empty set | A has no elements |
| ℘(A) | Power set | Set of all subsets of A |
5. Semantic Difference
| Aspect | Logic | Set Theory |
|---|---|---|
| Operates on | Propositions (T/F) | Sets and elements |
| Result | Boolean truth value | Set membership or set operations |