Applied Mathematics: Operation on Sets and Venn Diagrams
Category : 8th Class
Applied Mathematics: Operation on Sets and Venn Diagrams
Set
Set is a collection of well-defined objects which are distinct from each other. The objects in the set are called its elements. Sets are usually denoted by capital letters A, B, C, ....... and elements are usually denoted by small letters a, b, c, ........ For example, the set of all even natural numbers less than 10 can be represented by N = {2, 4, 6, 8}.
Methods for describing a set
(i) Roster Method: In this method, a set is described by listing elements, separated by commas, within braces, e.g. A = {a, e, i, o, u}
Note: This method is also called listing method or tabular form method.
(ii) Set builder method: In this method, we write down a rule which gives us all the elements of the set by that rule e.g. A = {x : x is a vowel of English alphabet}
Finite Set: A set containing finite number of elements or no element, is called a finite set.
e.g. The set of all persons in India is a finite set.
Infinite Set: A set containing infinite number of elements is called an infinite set.
Cardinality of a Finite Set: The number of elements in a given finite set is called cardinal number of finite set, denote by n (a), where A is the given set.
e.g. P = {5, 15, 25, 35, 45} \[\Rightarrow \] n (P) = 5
Empty Set (or null set)
A set containing no element in it, is called on empty set. It is represented by {} or \[\phi \] (read as phi), e.g. The set of all whole numbers less than o.
Singleton Set
A set containing a single element is called a singleton set. e.g. The set of ail prime numbers.
Equal Sets: Two sets A and B are called equal, if every element of A is a member of B and every element of B is a member of A. Thus we write A = B.
e.g. A = {2, 4, 6, 8, 10,} and {all the even natural numbers less than or equal to 10} ie. A and B are equal sets.
Find cardinal number of a set A of the composite numbers between 10 and 25.
(a) 4 (b) 6
(c) 8 (d) 9
(e) None of these
Answer (d)
Explanation: Here, A = {12, 14, 15, 16, 18, 20, 21, 22, 24}
\[\Rightarrow n\,(A)\,=\,9\,\Rightarrow \]Cardinal number of set A = 9
Disjoint Sets
Two sets A and B are called to be disjoint, if they have no elements in common. e.g. Sets A {2, 4, 6, 8, 10, 12} and B = {1, 3, 5, 7, 9, 11} \[\Rightarrow A\cap B\,\,=\phi \] \[\Rightarrow \]A and B are disjoint sets.
Subset and Superset
For any two sets A and B if every element of A is an element of B. then A is called the subset of B and B is called superset of A.
\[\Rightarrow A\subseteq B\,\,or\,\,B\supseteq A\]
Here the symbol \['\subset '\] stands for 'is a subset of' e.g. Set of all primes is a subset of the set of all whole numbers.
Note: Empty set is a subset of every set.
The number of subsets of a set A having n elements is\[{{2}^{n}}\].
Every set is a subset of itself ie. \[A\subset A\]and \[B\subset A.\]
If A = {3, 4, 5} and B = {1, 3, 4, 6, 5} then which one of the following in correct?
(a)\[A\subset B.\] (b) \[B\subset A.\]
(c) A = B (d) either (b) or (c)
(e) None of these
Answer (a)
Explanation: Since, all the elements of set A are contained in sets.
A is a subset of B but B is not a subset of A. So \[A\subset B.\]
Proper Subset
Let A and B are two sets, then A is called the proper subset of B if all elements of A are in B but B contains atleast one element that is not in A. Thus we can say that set A is not a proper subset of itself.
For example {1, 2} is a subset of {1, 2} but is not a proper subset of {1, 2}.
Note: If A is proper subset of B, then it is also a subset of B.
The proper subset of {1, 2, 3, 4, 5} is ________ .
(a) {1, 2, 3, 4} (b) {1, 2, 3, 4, 5}
(c) {1, 2, 3, 4, 5, 6} (d) All the above
(e) None of these
Answer (a)
Explanation: Clearly {1, 2, 3, 4} is a proper subset of {1, 2, 3, 4, 5}.
Universal Set
A set which consists of all the sets under discussion is called the universal set and is denoted by '\[\bigcup \]'.
For example: If A = {a, b, c}, B = {m, I, n} and C = {1, 4, 5} then U can be {a, b, c, m, I, n, 1, 4, 5}.
Complement of a Set
The set of all those elements of universal set which are not in set A is known as the complement of the set A. It is denoted by\[A'\text{ }or\text{ }{{A}^{c}}\].
If U is the universal set such that U = {3, 8, 12, 16} and A = {8, 12}, then find A'.
(a) {3, 8} (b) {8, 12}
(c) {3, 16} (d) {8, 12, 16}
(e) None of these
Answer (c)
Operations on Sets
1) Union of Sets: Let A and B be two sets. Then the union of A and B, represented by\[A\cup B\], is the set of all those elements which are either in A or in B or Both A and B ie. \[A\cup B\text{ }=\left\{ x\text{ }:\text{ }x\text{ }\in \text{ }A\text{ }and\text{ }x\text{ }\in \text{ }B \right\}.\]
Note: (i) \[A\cup U=U\,\,and\,\,A\cup \,\,\phi \,\,=\,\,A\]
(ii) If \[A\subseteq B,\text{ }then\text{ }A\cup B=\text{ }B\]
2) Intersection of Sets: Let A and B be two sets. Then the intersection of A and B, represented by\[A\cap B\], is the set of all those elements which are in both A and B. ie. \[A\cap B\text{ }=\left\{ x\text{ }:\text{ }x\text{ }\in \text{ }A\text{ }and\text{ }x\text{ }\in \text{ }B \right\}.\]
Note: (i) \[A\text{ }\cap \text{ }U\text{ }=A\text{ }and\text{ }A\cap \phi =\phi \]
(ii) If \[A\subseteq B,\text{ }then\text{ }A\cap B=A\]
(iii) For two disjoint sets A and B. A\[\cap \]B = \[\phi \]
3) Difference of Sets: Let A and B be two, sets, then the difference A-B is the set of all those elements which are in A but not in B ie. \[A\text{ }-\text{ }B\text{ }=\left\{ x\text{ }:\text{ }x\text{ }\in \text{ }A\text{ }and\text{ }x\text{ }\in \text{ }B \right\}.\]
Note: (i) if A ? B = B \[-\] A, then A = B
(ii) Fora set A, A' = U \[-\] A
If A = {1, 3, 4, 5, 1, 9} and {2, 4, 6, 8} then which one of the following is correct?
(a) A\[\cup \]B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(b) A\[\cap \]B = {4}
(c) A \[-\] B = {1, 3, 5, 7, 8}
(d) B \[-\] A = {2, 4, 6, 8}
(e) None of these
Answer (d)
Explanation: Clearly B \[-\] A = {2, 6, 8}
Venn Diagrams
The pictorial representation of sets by the means of diagram is called venn diagrams. In venn diagrams, the universal set is usually represented by a rectangular region. The subsets of universal set is represented by the closed region inside the rectangular region. The elements of the sets are shown in the closed regions.
If U = {1, 2, 3, 9, 10, 11, 13}, A = {2, 9, 6, 11} and B = {2, 9, 13, 1} then draw the venn diagram of the given information.
Solution:
Note: in the above venn diagrams, the shaded venn part clearly represented the diagrams A\[\cup \]B as:
(i) venn diagrams of A\[\cap \]B, as: (ii) venn diagrams of A-B, as:
Note: n (A \[\bigcup \] B) = n (A) + n (B) \[-\] n (A\[\cap \]B)
Ordered Pair
A pair of elements a, b arranged in a specific order (a, b) is called on ordered pair. i.e. if (a, b) = (5, 8) then a = 5 and b = 8
Note: (a, b) = (b, a) if and only if a = b
Cartesian product of set
If A and B are any two nonempty sets, then the cartesian product of A and B is represented as \[\left( A\times B \right)\]={(x, y): x\[\in \]A and y \[\in \] B}
Note: (i) We read \[\left( A\times B \right)\]as A cross B.
(ii) \[A\times B\ne B\times A\]
(iii) \[n\,(A\times B)=n\,(A)\cdot n\,(B)\]
(iv) \[A\,\Delta \,B=(A\bigcup B)-(A\cap B)\]
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