SETS
Frequently Asked Questions
A set is a well-defined collection of distinct objects called elements.
So that it is possible to clearly decide whether a given object belongs to the set or not.
The individual objects or members contained in a set are called its elements.
Sets are generally denoted by capital letters such as \(A,\, B,\, C\).
Elements are represented by small letters such as \(a, \,b,\, x\).
It means “belongs to” or “is an element of”.
It means “does not belong to” a given set.
A method of listing all elements of a set within curly braces.
A representation describing a set by a common property satisfied by its elements.
(\A = {2,4,6,8}\).
\(A = {x : x \text{ is an even natural number}}\).
A set containing no elements, denoted by \(\varnothing\).
Yes, there is only one empty set.
A set containing exactly one element.
A set with a definite number of elements.
A set with an unlimited number of elements.
A set containing all objects under consideration for a particular discussion.
It is usually denoted by \(U\).
A set \(A\) is a subset of \(B\) if every element of \(A\) is also an element of \(B\).
A subset that is not equal to the original set.
\(\subseteq\) represents subset, and \(\subset\) represents proper subset.
A set with \(n\) elements has \(2^n\) subsets.
The set of all subsets of a given set.
It is denoted by \(P(A)\).
The union of two sets contains all elements belonging to either or both sets.
The symbol for union is \(\cup\).
The intersection contains only those elements common to both sets.
The symbol for intersection is \(\cap\).
Sets having no common elements.
The set of all elements in the universal set not belonging to the given set.
The complement of \(A\) is written as \(A^'\) or \(A^c\).
\(\varnothing\) has no elements, while \({\varnothing}\) has one element.
\(A \cup B = B \cup A\).
\(A \cap B = B \cap A\).
\((A \cup B) \cup C = A \cup (B \cup C)\).
\((A \cap B) \cap C = A \cap (B \cap C)\).
\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\).
\((A \cup B)' = A' \cap B'\).
\((A \cap B)' = A' \cup B'\).
Diagrams using closed curves to represent sets and their relationships visually.
They simplify understanding of set operations and relationships.
The number of elements in a set.
It is denoted by \(n(A)\).
\(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).
A method to calculate the number of elements in unions of sets accurately.
No, as per standard definition, a set does not contain itself.
No, all elements of a set must be distinct.
Conceptual definitions, Venn diagram problems, formulas, and numerical applications.
It provides the basic language and structure for all other topics in mathematics.
Sets are used in data classification, logic, probability, and computer science.
