Worksheet III

 SETS

Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A, B, C , X , Y , Z etc. and elements of a set are denoted by a, b, c , x , y, z etc.

If a is an element of set A, then we say that a belongs to A. The phrase ‘belongs to’ denoted by the Greek symbol Î (epsilon). Thus, we write and written as a Î A and b does not belongs to set A is written as b Ï A.

 Representation of Sets

There are two ways of representing a set:

(i)      Roster form or Tabular form or Listing method : In the roster form, we list all the elements of the set within curly braces { } and separate them by commas.

(ii)Set-builder form or Rule method : In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

 Types of Sets

1.     Empty set: A set which does not contain any element is called an empty set or the void set or the null set and it is denoted by {   } or j .

2.     Singleton set: A set consisting of a single element, is called a singleton set.

3.     Finite and infinite: sets A set which is empty or consists of a finite number of elements is called a finite set, otherwise, the set is called an infinite set.

4.     Equivalent sets: Two finite sets A and B are said to be equal, if they have equal number of elements, i.e. n( A) = n ( B ).

5.     Equal sets: Two sets A and B are said to be equal, if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ¹ B.

 Subset

A set A is said to be a subset of a set B, if every element of A is also an element of B. In symbols, we can write

A Ì B ,  if x Î A Þ x Î B

Also, if A Ì B and A ¹ B, then A is called a proper subset of B and B is called superset of A.

Note

(i)       Every set is a subset of itself.

(ii)       The empty set is a subset of every sets.

(iii)The total number of subsets of a finite set containing n elements is 2ⁿ .

 Power Set

The collection of all subsets of a set A is called the power set of A. It is denoted by P ( A). If the number of elements in A, i.e. n( A) = m,  then the number of elements in P ( A) i.e. n [P ( A)] = 2^m .

Properties of Power Sets

(i)  If A Í B, then P ( A) Í P ( B ).

(ii)  P ( A) Ç P ( B ) = P ( A Ç B )

(iii)  P ( A È B ) ¹ P ( A) È P ( B )

Universal Set

If there are some sets under consideration, then there happens to be a set which is a superset of each one of the given sets. Such a set is known as the universal set and is denoted by U.

Intervals as Subsets of R

1.     The set of real numbers {x : a < x < b} is called an open interval and is denoted by (a, b ).

2.     The set of real numbers {x : a £ x £ b} is called a closed interval and is denoted by [a, b].

3.     The intervals closed at one end and open at the other are known as semi-open or semi-closed intervals. [ a, b ) = {x : a £ x < b} is an open interval from a to b which includes a but excludes b. (a , b ] = {x : a < x £ b} is an open interval from a to b which excludes a but includes b.

 Venn Diagrams

Venn diagrams are the diagrams, which represent the relationship between sets. In Venn diagrams, the universal set is represented usually by a rectangular region and its subset are represented usually by circle or a closed geometrical figure inside the universal set. Also, an element of a set is represented by a point within the circle of set.

Example. If U = {1, 2, 3, 4, ..., 10} and A = {1, 2, 3}, then its Venn diagram is as shown in the figure

 Operations on Sets

(i)  Union of sets : The union of two sets A and B is the set of all those elements which belong to either in A or in B or in both A and B. It is denoted by A È B.

Thus, A È B = {x : x Î A or x Î B}

(ii)  Intersection of sets : The intersection of two sets A and B is the set of all those elements, which are common to both A and B. It is denoted by A Ç B.

Thus, A Ç B = {x : x Î A and x Î B}

(iii)  Disjoint sets : Two sets A and B are said to be disjoint sets, if they have no common elements

i.e. if A Ç B = j.

Laws of Algebra of Sets

(i)  Idempotent laws For any set A, we have

(a) A È A = A                  (b) A Ç A = A

(ii)  Identity laws For any set A, we have

(a)   A È j = A                    (b) A Ç U = A

(iii)  Commutative laws For any two sets A and B, we have

(a)A È B = B È A             (b) A Ç B = B Ç A

(iv)  Associative laws For any three sets A, B and C , we have

 (a) A È ( B È C ) = ( A È B ) È C

(b) A Ç ( B Ç C ) = ( A Ç B ) Ç C

(v)  Distributive laws If A, B and C are three sets, then

(a) A È ( B Ç C ) = ( A È B ) Ç ( A È C )

(b) A Ç ( B È C ) = ( A Ç B ) È ( A Ç C )

MCQ Type Questions

Question 1.
If A, B and C are any three sets, then A – (B ∪ C) is equal to
(a) (A – B) ∪ (A – C)
(b) (A – B) ∪ C
(c) (A – B) ∩ C
(d) (A – B) ∩ (A – C)

Question 2.
(A’)’ = ?
(a) ∪ – A
(b) A’
(c) ∪
(d) A

Question 3.
A – B is read as?
(a) Difference of A and B of B and A
(b) None of the above
(c) Difference of B and A
(d) Both a and b

Question 4.
If A, B and C are any three sets, then A × (B ∪ C) is equal to
(a) (A × B) ∪ (A × C)
(b) (A ∪ B) × (A ∪ C)
(c) None of these
(d) (A × B) ∩ (A × C)

Question 5.
IF A = [5, 6, 7] and B = [7, 8, 9] then A ∪ B is equal to
(a) [5, 6, 7, 8, 9]
(b) [5, 6, 7]
(c) [7, 8, 9]
(d) None of these

Question 6.
Which of the following sets are null sets
(a) {x: |x |< -4, x ?N}
(b) 2 and 3
(c) Set of all prime numbers between 15 and 19
(d) {x: x < 5, x > 6}

IF R = {(2, 1),(4, 3),(4, 5)}, then range of the function is?
(a) Range R = {2, 4}
(b) Range R = {1, 3, 5}
(c) Range R = {2, 3, 4, 5}
(d) Range R {1, 1, 4, 5}

Question 8.
The members of the set S = {x | x is the square of an integer and x < 100} is
(a) {0, 2, 4, 5, 9, 58, 49, 56, 99, 12}
(b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
(c) {1, 4, 9, 16, 25, 36, 64, 81, 85, 99}
(d) {0, 1, 4, 9, 16, 25, 36, 49, 64, 121}

Question 9.
In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?
(a) 19
(b) 41
(c) 21
(d) 57

Question 10.
{ (A, B) : A² +B² = 1} on the sets has the following relation
(a) reflexive
(b) symmetric
(c) none
(d) reflexive and transitive

Question 11.
Two finite sets have N and M elements. The number of elements in the power set of first set is 48 more than the total number of elements in power set of the second test. Then the value of M and N are
(a) 7, 6
(b) 6, 4
(c) 7, 4
(d) 6, 3

Question 12.
The range of the function f(x) = 3x – 2‚ is
(a) (- ∞, ∞)
(b) R – {3}
(c) (- ∞, 0)
(d) (0, – ∞)

Question 13.
If A, B, C be three sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then,
(a) B = C
(b) A = C
(c) A = B = C
(d) A = B

Question 14.
In 2nd quadrant?
(a) X < 0, Y < 0
(b) X < 0, Y > 0
(c) X > 0, Y > 0
(d) X > 0, Y < 0

Question 15.
Empty set is a?
(a) Finite Set
(b) Invalid Set
(c) None of the above
(d) Infinite Set

Question 17.
If A = [5, 6, 7] and B = [7, 8, 9] then A U B is equal to
(a) [5, 6, 7, 8, 9]
(b) [5, 6, 7]
(c) [7, 8, 9]
(d) None of these

Question 18.
Which of the following two sets are equal?
(a) A = {1, 2} and B = {1}
(b) A = {1, 2} and B = {1, 2, 3}
(c) A = {1, 2, 3} and B = {2, 1, 3}
(d) A = {1, 2, 4} and B = {1, 2, 3}

Question 19.
In a class of 50 students, 10 did not opt for math, 15 did not opt for science and 2 did not opt for either. How many students of the class opted for both math and science.
(a) 24
(b) 25
(c) 26
(d) 27

Question 20.
In last quadrant?
(a) X < 0, Y > 0
(b) X < 0, Y < 0
(c) X > 0, Y < 0
(d) X > 0, Y > 0