SET theory | concept | questions

Definition:

A Set is a collection of well-defined objects or elements. The term well-defined stands for definition assigns it a unique interpretation or value.

For example the collection of all the vowels in English alphabets is a set. where as the collection of Tall tree is not a set. As the term “Tall” is not well defined.

Representation of a Set:

There are two methods of representation of a set:
(1) Roster or Tabular form: In this method we describe a set by listing the elements, separated by commas within curly brackets.
(2) Set builder form: Here all the elements of a set possess a single common property which is not possess by any other elements out side the set.

Types of Sets:

Empty Set: A Set which does not contains any elements is called the empty set or the null set or the void set. The empty set is denoted by the symbol $\phi$ or $\{\}$
Finite and Infinite Set: A Set which is empty or consists of a definite number of elements is called finite set, otherwise the set is called infinite set.
Singleton Set: A set consists of only one element.
Subset: Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B and we write . Here B is called superset of A and also written as
- Every set is a subset of itself.
- The empty set is the subset of every set.

Equal Sets: Two sets A and B are said to be equal if every element of A are also an element of B and every element of the set B are also in A. In other words $A \subseteq B$ and $B \subseteq A$ implies $A=B$ .
If A is a set with $n (A) = m$ , then the number of subsets of A are $2^m$ .
Power Set: The set of all subsets of a given set A is called the power set of A and is denoted by P(A).
- For example, if A = {1, 2, 3}, then P(A) = { $\phi$ , {1}, {2}, {3}, {1, 2} {1, 3}, {2, 3}, {1, 2, 3}}
- if A has n elements, then its power set P(A) contains exactly $2^n$ elements
Universal Set: A Set U that contains all sets in a given context is called the universal set.

Operations on Sets

Union of Two Sets: The union of two sets A and B, written as $A \cup B$ (read as ‘A union B’), is the set consisting of all the elements which are either in A or in B or in both. i.e., $A \cup B=\{x: x \in A$ or $x \in B\}$
Intersection of Two Sets: The intersection of two sets A and B, written as $A \cap B$ (read as ‘A intersection B’), is the set consisting of all the elements which are both in A and B. i.e., $A \cap B=\{x: x \in A$ and $x \in B\}$
Difference of sets: The difference of two sets A and B is the set of all the elements of A which do not belongs to B. $A - B=\{x: x \in A$ and $x \notin B\}$
Symmetric difference of Two Sets: The symmetric difference of two sets A and B, is
- - $A \Delta B= (A \cup B) - (A \cap B)$
  - $A \Delta B= (A - B) \cup (B-A)$

Complement of a Sets: The complement of a set A, written as $A'$ or $A^C$ , is the set consisting of all the elements which are not in A. i.e., $A '=\{x: x \in U$ and $x \notin A\}$

Algebra of Set operation

Idempotent Law: $A \cup A=A$ $A \cap A=A$
Identity Law: For any set A, $A \cup \phi=A$ $A \cap U=A$
Commutative Law: $A \cup B=B \cup A$ $A \cap B=B \cap A$
Associative Law: For any three sets A, B, C
$A \cup (B \cup C)= (A \cup B) \cup C$
$A \cap (B \cap C)= (A \cap B) \cap C$
Distributive Law: For any three sets A, B, C
$A \cup (B \cap C)= (A \cup B) \cap ((A \cup C)$
$A \cap (B \cup C)= (A \cap B) \cup ((A \cap C)$
De-morgan’s Law: For any two sets A and B
$(A \cup B)'= A' \cap B'$
$(A \cap B)'= A' \cup B'$
Cardinality or Order of a finite set: The cardinality of a finite set is the number of elements in the set. The cardinality is denoted by $|A|$ .
$|A \cup B|=|A|+|B|- |A \cap B|$
$(|A \cup B\cup C|=|A|+|B|+|C|- |A \cap B|- |B \cap C|- |A \cap C|+-|A \cap B\cap C|$
$|A \Delta B|=|A-B|+|B-A|=|A|+|B|-2 |A \cap B|$

For more understanding of the concept you can view some of the solved problems and MCQs regarding Set theory