SET

In mathematics, a set is a well-defined collection of distinct elements.

Sets are conventionally denoted with capital letters such as A, B, C.

The elements of a set are defined with lowercase letters such as a, b, c.

= : equal sign; equality.

≠ : not equal sign; inequality.

∈ : element of; belongs to; set membership.

∉ : not element of; no set membership.

a ∈ A : "a is an element of A" or "a belongs to A".

∃ : exists

∄ : does not exist

! : unique

∃! : exists and it is unique

⇒ : implies

P ⇒ Q : P implies Q; if P is true implies that Q is true.

⇔ : equivalent; if and only if (iff).

P ⇔ Q : if P is true then Q is true and vice versa.

⊥ : contradiction

↯ : contradiction

P:= {n ∈ N | n is a prime number} : representation of a set by property.

⊆ : subset

A ⊆ B : subset; A is a subset of B; set A is included in set B; {9,14,28} ⊆ {9,14,28}

⊂ : proper subset

A ⊂ B : proper subset / strict subset; A is a subset of B, but A is not equal to B; {9,14} ⊂ {9,14,28}

⊄ : not subset

A ⊄ B : not subset; set A is not a subset of set B; {9,66} ⊄ {9,14,28}

⊇ : superset

A ⊇ B : superset; A is a superset of B; set A includes set B; {9,14,28} ⊇ {9,14,28}

A ⊃ B : proper superset / strict superset; A is a superset of B, but B is not equal to A; {9,14,28} ⊃ {9,14}

⊅ : not superset

A ⊅ B : not superset; set A is not a superset of set B; {9,14,28} ⊅ {9,66}

A = {1,3,7,a} : representation of a set by enumeration.

|A| : cardinality; the number of elements of set A.

#A : cardinality; the number of elements of set A.

Sets can be represented with diagrams of Euler and Venn.

David Hilbert (1862 - 1943) was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries.

Bertrand Russell (1872 - 1970) was a British polymath, philosopher, logician, mathematician, historian, writer, social critic, political activist, and Nobel laureate.

Russell's paradox: A = {X | X ∉ X}; A ∈ A ⇒ A ∉ A : ↯; A ∉ A ⇒ A ∈ A : ↯

Kurt Gödel (1906 - 1978) was a German-Austrian logician, mathematician, and analytic philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history.

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

{ } : set; a collection of elements; A = {3,7,9,14}, B = {9,14,28}

∪ : union

A ∪ B : union; objects that belong to set A or set B; A ∪ B = {3,7,9,14,28}

A ∪ B = {c | c ∈ A or c ∈ B}

∩ : intersection

A ∩ B : intersection; objects that belong to set A and set B; A ∩ B = {9,14}

A ∩ B = {c | c ∈ A and c ∈ B}

A \ B : relative complement; objects that belong to A and not to B; A = {3,9,14}, B = {1,2,3}, A \ B = {9,14}

A - B : relative complement; objects that belong to A and not to B; A = {3,9,14}, B = {1,2,3}, A - B = {9,14}

A \ B = {a ∈ A | a ∉ B}

A × B : cartesian product; set of all ordered pairs from A and B; A × B = {(a,b) | a ∈ A , b ∈ B}; (a,b) ≠ (b,a)

A ∪ B = B ∪ A : on the union of sets the commutative property is valid.

A ∩ B = B ∩ A : on the intersection of sets the commutative property is valid.

A \ B ≠ B \ A : on the difference of sets the commutative property is not valid.

A x B ≠ B x A : on the cartesian product of sets the commutative property is not valid.

A = {1,2,3}, B = {3,5}; A ∪ B = {1,2,3,5}; A ∩ B = {3}; A \ B = {1,2}; B \ A = {5}; A x B = {(1,3),(1,5),(2,3),(2,5),(3,3),(3,5)}

Card(A) = n, Card(B) = m, Card(A x B) = nm

A x A = A^{2}

A x B x C = {(a,b,c) | a ∈ A, b ∈ B, c ∈ C}

A x A x ... x A = A^{n}