Languauge and notations

Common Symbols & Their Meanings

∈

∈: 'is an element of' or 'belongs to'. It indicates that an element is a member of a set. For example, if A=1,2,3A, then 2∈A means 2 is an element of set A.

∉

∉: 'is not an element of'. It indicates that an element is not a member of a set. For example, if A=1,2,3 then 4∉A means 4 is not an element of set A.

⊆

⊆: 'is a subset of' or 'contains all elements of'. It denotes that every element of one set is also in another set, such as if B={1,2} and A={1,2,3}, then B⊆A means B is a subset of A.

⊂

⊂: 'is a proper subset of'. It indicates that one set is a subset of another, but not equal to it. For example, if A={1,2,3} and B={1,2}, then B⊂A means B is a proper subset of A.

∅ or { }

represents the empty set, which contains no elements. It indicates the absence of any members in a set. For example, if A=∅, then A has no elements.

ℕ

denotes the set of natural numbers, which includes all positive integers starting from 1, i.e., ℕ = {1, 2, 3, …}. It may or may not include zero, depending on the definition used.

ℤ

denotes the set of integers, which includes all whole numbers, both positive and negative, as well as zero, i.e., ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}.

ℚ

denotes the set of rational numbers, which include all numbers that can be expressed as a fraction of two integers, where the denominator is not zero, i.e., ℚ = {p/q | p, q ∈ ℤ, q ≠ 0}. It includes integers, fractions, and can be finite or infinite.

| or :

denotes absolute value or is used in set notation to indicate "such that".

∩

∩: denotes the intersection of two sets, representing elements that are common to both sets. For instance, if set A contains {1, 2, 3} and set B contains {2, 3, 4}, then A∩B=2,3, because 2 and 3 are the elements present in both sets A and B. The intersection focuses only on shared elements.

∪

denotes the union of two sets, representing all elements that are in either set. For example, if set A contains {1, 2, 3} and set B contains {3, 4, 5}, then A∪B={1, 2, 3, 4, 5}, including all unique elements from both sets.

A′

A′ denotes the complement of set A, which includes all elements not in set A but are within the universal set U. For example, if the universal set is U = {1, 2, 3, 4, 5} and set A contains {1, 2, 3}, then A′ = {4, 5}, because 4 and 5 are in U but not in A. More formally, A′= x∣x ∈ U , x ∉ A′, where U is the universal set.

What is a proper subset

A proper subset refers to a set that contains some but not all elements of another set, meaning it has at least one element in common with the original set, but is not identical to it. For example, if set A = {1, 2, 3}, then B = {1, 2} is a proper subset of A.

What is a subset

A subset is a set where all elements are contained within another set, which means every element of the subset is also an element of the original set. For example, if set A = {1, 2, 3}, then B = {2} is a subset of A.