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Unit 01: SETS
INTRODUCTION TO CHAPTER
Explanation Of Exercise 1.2
OPERATIONS ON TWO SETSTwo sets can be combined in many different ways.
1. Union of Two Sets:
Let A and B be sets. The union of the sets A and B, denoted by AUB, is the set that contains those elements which are contained in A or B or both.
AUB = { x | x ∈ A ∨ x ∈ B }
For example:
If A = {1, 2, 5, 8} and B = {1, 3, 5, 9},
then AUB = {1, 2, 3, 5, 8, 9}
2. Intersection of Two Sets:
Let A and B be sets. The intersection of sets A and B, denoted by A ∩ B, is the set that contains those elements which are contained in both A and B.
A ∩ B = { x | x ∈ A ∧ x ∈ B }
For example:
If A= {a, b, c, d} and B = {b, d, e, f},
then A ∩ B = {b, d}.
3. Difference of Two Sets:
Let A and B be sets. The difference of sets A and B, denoted by A - B, is the set containing those elements that are in A but not in B.
A - B = { x | x ∈ A ∧ x ∉ B }
For example:
If A = {1, 2, 3, 4, 5} and B {2, 4, 6, 8},
then A - B = {1, 3, 5} and B - A = {6, 8}.
4. Symmetric Difference Of Two Sets:
Let A and B be sets. The symmetric difference of sets A and B, denoted by A ∆ B, is the set containing those elements which are either in A or in B but not in both A and B.
A ∆ B = (AUB) - (A ∩ B)
Example No. 1:
Find A ∆ B, if A = {1, 2, 3, 4, 6} and B = {1, 3, 5, 7}
Solution:
Since A = {1, 2, 3, 4, 6} and
B = {1, 3, 5, 7},
As A ∆ B = A U B - A ∩ B
A ∆ B = {1, 2, 3, 4, 5 , 6, 7} - {1, 3}
therefore A ∆ B = {2, 4, 5, 6, 7} Ans.
5. Universal Set:
A set which contains all the sets under consideration is called a Universal Set. Usually it is denoted by U.
For example:
- The set of all students in your school is a Universal Set U.
-
Subset Of universal sets:
Different sets of students of all classes in the school such as students of 9th class and students of 10th class, etc., are all subsets of the set U of all the students of the school.
6. Complement of a Set:
Let U be a universal set and let A ⊂ U. The complement of set A, denoted by Ac or A', is the set containing those elements of U, which are not in A.
For Example:
If U = {1, 2, 3, . ,10} and A = {1, 3, 5, 7, 9), then
A' = U - A = {2, 4, 6, 8, 10} .
Also (A')' = A
OPERATIONS OF UNION AND INTERSECTION ON THREE SETS
If A, B and C are any three sets, then the following operations of union and intersection can be performed.
- A U (B U C)
- (A U B) U C
- A ∩ (B ∩ C)
- (A ∩ B) ∩ C
- A U (B ∩ C)
- A ∩ (B U C)
- (A U B) ∩ C
- (A ∩ B) U C
- (A U B) ∩ (A U C)
- (A ∩ B) U (A ∩ C)
A few of these operations are explained by the following examples:
Suppose A= {a, b, c}, B= {b, c, d, e}, C= {c, d, e, f, h}
(i) AU(BUC)
Solution: = {a, b, c} U ({b, c, d, e} U {c, d, e, f, h})
= {a, b, c} U {b,c,d,e,f,h}
So, AU(BUC) = {a, b, c, d, e, f, h}
(ii) (AUB)UC
Solution:
= ({a, b, c} U {b, c, d, e}) U {c, d, e, f, h}
= (a, b, c, d, e} U (c, d, e, f, h)
So, (AUB)UC = {a, b, c, d, e, f, h}
(v) AU(B∩C)
Solution:
= {a, b, c} U ({b, c, d,e) ∩ {c, d, e, f, h)
= {a, b, c} U (c, d, e)
So, AU(B∩C) = {a, b, c d, e}
(ix) (AUB) ∩ (AUC)
Solution:
= ( {a, b, c} U {b, c, d, e}) ∩ ({a, b, c} U (c, d, e, f, h})
= {a, b, d, c, d, e} ∩ {a, b, c, d, e. f. h}
So, (AUB) ∩ (AUC) = {a, b, c, d, e}
FUNDAMENTAL PROPERTIES OF UNION AND INTERSECTION FOR TWO OR THREE SETS:
(1) Commutative Property of Union: (For any two sets A and B)
A U B = B U A
For example:If A = {a} and B = (a, b} then
A U B = {a} U {a, b} = {a, b} and
B U A = {a, b} U {a} = {a, b}
Hence A U B = B U A
(2) Commutative Property of Intersection: (For any two sets A and B)
A ∩ B = B ∩ A
For example:If A = {a} and B = (a, b} then
A ∩ B = {a} ∩ {a, b} = {a} and
B ∩ A = {a, b} U {a} = {a, b}
Hence A ∩ B = B ∩ A
(3) Associative Property of Union: (For any three sets A, B and C)
A U (B U C) = (A U B) U C
For Example:If A = {a}, B = {a, b} C = {a, b, c}, then
A U (B U C) = {a} U ({a, b} U {a, b, c}) = {a} U {a, b, c} = {a, b, c} and
(A U B) U C = ({a} U {a, b}) U {a, b, c} = {a, b, c}
Hence A U (B U C) = (A U B) U C
(iv) Associative Property of Intersection: (For any three sets A, B and C)
A ∩ (B ∩ C) = (A ∩ B) ∩ C
For example:If A = {a}, B = {a, b}, C = {a, b, c} then
A ∩ (B ∩ C) = {a} ∩ {(a, b) ∩ (a, b, c)} = {a} ∩ {a, b} = {a} and
(A ∩ B) ∩ C = {(a} ∩ {a, b)} ∩ {a, b, c} = {a} ∩ {a, b, c} = {a}
Hence A ∩ (B ∩ C) = (A ∩ B) ∩ C
(v) Distributive Property of Union over Intersection: (For any three sets A, B and C)
A U (B ∩ C) = (A U B) ∩ (A U C)
(vi) Distributive Property of Intersection over Union: (For any three sets A, B and C)
A ∩ (B U C) = (A ∩ B) U (A ∩ C)
De Morgan's Laws:
Let U be a Universal Set and A, B be any subsets of U. Then
(i) (AUB)' = A'∩B'
(ii) (A∩B)' = A'UB'
These laws are known as De Morgan's Laws.
Venn Diagrams:
Sets can also be represented graphically using Venn diagrams named after the English Mathematician John Venn, who introduced their use in 1881.
In Venn diagrams the universal set U, which contains all the elements of the subsets under consideration, is usually represented by a rectangle. Inside this rectangle, circle or other simple closed geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set. Venn diagrams are often used to indicate the relationships between the sets.
Example:
Draw a Venn diagram to represent
U = {1, 2, 3,....,10} and A = {1, 3, 5, 7, 9}
Solution:
We draw a rectangle to indicate the universal set U. Inside this rectangle we draw a circle to represent A. Inside this circle we indicate the elements of A with points as shown below:
Explanation:
Venn diagrams can be used to show that a set A is a subset of a set B. We draw the universal set U as a rectangle. Within this rectangle we draw a circle for B. Since A is a subset of B, we draw another circle for A within the circle for B. This relationship is shown below:
In the case of operations with more than two sets, the Venn diagrams help us in proving equalities
(AUB)UC = AU(BUC) and (A∩B)∩C = A∩(B∩C)
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