Discrete Structures(LECTURE 2)

                                                    MARWARI COLLEGE RANCHI 

                                    (UNDER RANCHI UNIVERSITY ,RANCHI)

NAME : RAJU MANJHI

SEM: II (B.Sc(CA)/IT)

SUB: Discrete Structures

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4. Proper Subset: If A is a subset of B and A ≠ B then A is said to be a proper subset of B. If A is a proper subset of B then B is not a subset of A, i.e., there is at least one element in B which is not in A.

Example:

(i) Let A = {2, 3, 4}
B = {2, 3, 4, 5}

A is a proper subset of B.

(ii) The null ∅ is a proper subset of every set.

5. Improper Subset: If A is a subset of B and A = B, then A is said to be an improper subset of B.

Example

(i) A = {2, 3, 4}, B = {2, 3, 4}

A is an improper subset of B.

(ii) Every set is an improper subset of itself.

6. Universal Set: If all the sets under investigations are subsets of a fixed set U, then the set U is called Universal Set.

Example: In the human population studies the universal set consists of all the people in the world.

7. Null Set or Empty Set: A set having no elements is called a Null set or void set. It is denoted by∅.

8. Singleton Set: It contains only one element. It is denoted by {s}.

Example: S= {x|x∈N, 7<x<9} = {8}

9. Equal Sets: Two sets A and B are said to be equal and written as A = B if both have the same elements. Therefore, every element which belongs to A is also an element of the set B and every element which belongs to the set B is also an element of the set A.

                         A = B ⟺ {x ϵ A  ⟺  x ϵ B}.

If there is some element in set A that does not belong to set B or vice versa then A ≠ B, i.e., A is not equal to B.

10. Equivalent Sets: If the cardinalities of two sets are equal, they are called equivalent sets.

Example: If A= {1, 2, 6} and B= {16, 17, 22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3

11. Disjoint Sets: Two sets A and B are said to be disjoint if no element of A is in B and no element of B is in A.

Example:

R = {a, b, c}
S = {k, p, m}

R and S are disjoint sets.

12. Power Sets: The power of any given set A is the set of all subsets of A and is denoted by P (A). If A has n elements, then P (A) has 2ⁿ elements.

Example: A = {1, 2, 3}
P (A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

 

Operations on Sets

The basic set operations are:

1.    Union of Sets: Union of Sets A and B is defined to be the set of all those elements which belong to A or B or both and is denoted by A∪B.

 

                 A∪B = {x: x ∈ A or x ∈ B} 

 

Ex: Let A = {1, 2, 3},       B= {3, 4, 5, 6}
        A∪B = {1, 2, 3, 4, 5, 6}. 

 

2.    Intersection of Sets: Intersection of two sets A and B is the set of all those elements which belong to both A and B and is denoted by A ∩ B.

    A ∩ B = {x: x ∈ A and x ∈ B}  

 

                  Ex: Let A = {11, 12, 13},       B = {13, 14, 15}
                                  A ∩ B = {13}.

 

3.    Difference of Sets: The difference of two sets A and B is a set of all those elements which belongs to A but do not belong to B and is denoted by A - B.

   A - B = {x: x ∈ A and x ∉ B}  

Ex: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A - B = {3, 4} and B - A = {5, 6}

 

 

 

4.    Complement of a Set: The Complement of a Set A is a set of all those elements of the universal set which do not belong to A and is denoted by A^c.

                     A^c = U - A = {x: x ∈ U and x ∉ A} = {x: x ∉ A}

 

Ex: Let U is the set of all natural numbers.
A = {1, 2, 3}
A^c = {all natural numbers except 1, 2, and 3}.