Frequently Used Mathematical Symbols:
- Equality: =
- Used to show that two expressions are equivalent. Example: 2 + 2 = 4.
- Inequality: <, >, <=, >=
- Less than, greater than, less than or equal to, greater than or equal to. Example: 3 < 5, x >= 0.
- Logical Connectives:
- And: ∧ or & in text form. Example: P ∧ Q (P and Q are both true).
- Or: ∨ or | in text form. Example: P ∨ Q (at least one of P or Q is true).
- Not: ¬ or ~ in text form. Example: ¬P (P is not true).
- Implies: → or => in text form. Example: P → Q (if P is true, then Q must be true).
- If and only if: <-> or <=> in text form. Example: P <-> Q (P is true if and only if Q is true).
- Quantifiers:
- For all: ∀ (for all). Example: ∀x (P(x)) (the property P holds for all x).
- There exists: ∃ (there exists). Example: ∃x (P(x)) (there is at least one x for which P holds).
- Set Theory:
- Element of: ∈. Example: x ∈ A (x is an element of set A).
- Subset: ⊆. Example: A ⊆ B (A is a subset of B).
- Union: ∪. Example: A ∪ B (the set of all elements in either A or B or both).
- Intersection: ∩. Example: A ∩ B (the set of elements that are in both A and B).
- Infinity: ∞ (infinity, often used in philosophical discussions about sets or potentiality).
Basic Axioms and Principles:
- Axiom of Extensionality: If two sets have exactly the same elements, then they are the same set. Text form: ∀A ∀B [ (∀x (x ∈ A <-> x ∈ B)) -> A = B ]
- Axiom of Pairing: For any two sets A and B, there exists a set C whose elements are exactly A and B. Text form: ∀A ∀B ∃C ∀x [ x ∈ C <-> (x = A ∨ x = B) ]
- Axiom of Infinity: There exists a set containing the empty set and for every element in the set, there's another element which is the union of the element with itself. This ensures the existence of an infinite set. Text form: ∃S [ ∅ ∈ S ∧ ∀x ∈ S ∃y ∈ S (x ∪ {x} = y) ]
- Axiom of Power Set: For any set A, there exists a set P(A) which contains all subsets of A. Text form: ∀A ∃P(A) ∀B [ B ∈ P(A) <-> ∀x (x ∈ B -> x ∈ A) ]
- Axiom of Choice: For any set S of non-empty sets, there exists a choice function f that selects one element from each set in S. This is often debated in philosophical contexts for its implications on determinism vs. freedom. Text form: ∀S [ (∀A ∈ S (A ≠ ∅)) -> ∃f (f is a function ∧ ∀A ∈ S (f(A) ∈ A)) ]
These symbols and axioms are fundamental in mathematical philosophy, particularly when discussing logic, set theory, and the foundations of mathematics. They help in formulating arguments about what exists, how we can know it, and what we can say about it logically.
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