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Sunday, 15 January 2017

Discrete Mathematics - Sets and Types Of Sets

Sets and Types Of Sets

Discrete Mathematics - Sets and Types Of Sets

Question: What is a Set?

Answer: A collection of well defined objects are called set. Upper case letters are used to denote a set.
Some examples of set are
(i) A = {I, 3, 5, 7, ... ,99},
(ii) B = {I, 4, 9,16,25, ... },
(iii) B = {x : x is a real number satisfying x(x²-4) = O}


Question: Identify which of the following are examples of set?

Collection of all chairs and tables of a class room
All tall men of a class
All students of BCA class
All good students of a college.
Collection of Indian Prime Ministers of age more than 150 years.

Answer: ⒜, ⒞, ⒠ are examples of set. ⒠ is a null set.


Question: What is a Null set or empty set?

Answer: A set having no element is called null set or empty set.
It is generally denoted by the symbol ∅.



For example:
∅ = {x : x is a real number satisfying x²+ 4 = 0}


Question: What is a Singleton set or unit set?

Answer: A set having only one element is called a unit set or a singleton.
e.g. S = {9}


Question: What is a pair set?

Answer: A set which contains only two elements is known as a pair set. Consider the examples
S = {e, f}
S = {{a}, {1, 3, 5}}


Question: Define Finite set and infinite set:

Answer: A set is called finite if it be empty or contains a finite number of elements, otherwise a set is called infinite.

Example of Finite set: A set containing even integers between 2 and 20
Example of Infinite set: A set containing even integers.


Question: What is a sub-set? What is a super-set?

Answer: If every element of a set A be also an element of another set B, then A is called a sub-set of Band B is called super set of A.
Symbolically, A ⊆ B or B ⊇ A.

If B contains some elements which are not elements of a A, then A is a proper sub-set of B. Or B is proper super-set of A.
i.e. A ⊂ B or B ⊃ A


Question: What are Disjoint sets?

Answer: Two sets A and B are called disjoint if they have no element in common.
For example, A = {1, 2, 3} and B = {10, 11, 12} are disjoint.


Question: When two sets are said to be identical or equal?

Answer: Two sets A and B are said to be equal or identical, i.e., A = B, if they have the same elements. In this case, all elements of A are also the elements of B.

For example, A = {I, 2, 3} and B == {3, 1, 2} are equal.


Question: What is a Universal set?

Answer: A universal set is the set of all elements from which elements may be chosen to form sets for a particular problem. It is generally denoted by S or U.

e.g.
U = {a, b, c, d, e, ..., z}


Question: What do you mean by Cardinality of a set?

Answer: If S be a set, then the number of elements present in the set S is known as cardinality of S and is denoted by |S|. Mathematically if S = {s1, s2, s3, ..., sk}, then |S| = k; k ∈ N.

For example, A = {2, 4, 8, 16, 32, 64, 128, 256}
∴  |A| = 8


Question: When the two sets (say A and B) are said to be comparable?

Answer: Two sets A and B are said to be comparable if any one of the following relation holds.
i.e.
(i) A ⊂ B or
(ii) B ⊂ A or
(iii) A = B.


Question: What is power set?

Answer: If A be a set, then the set of all subsets of A is known as power set of A. Which is denoted
by P(A).
Mathematically, P(A) = {X : X ⊆ A}

e.g.
A = {1, 2}
P(A) = {{1}, {2}, {1, 2}, ∅}

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