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SETS

Consider the following examples of different items.

1. Bat, Ball, Gloves, Wickets, Pads

2. Bag, Text Books, Note Books, Pen, Pencil, Eraser, Sharpener, Lunch Box, Water Bottle

3. a, e, i, o, u

 All these examples are the collections of items/ objects. Each example has something in common. In first example we see things of a cricket kit, In second example things that are carried to school. Third example vowels. These are well defined collection of objects or ideas.

 A well - defined collection of objects or ideas is known as a set.

 The meaning of the phrase "Well - defined" is

   (i) All the objects in the set should have a common feature or property.

   (ii) It should be possible to decide whether any given object belongs to the set or not.

Consider the example.

   "The collection of 5 good books in the school library".

   As the words "good books" are subjective in nature. Each student have their own collection of 5 books. A student who is interested in comics picks, comics books. A student who is interested in maths may pick 5 Maths related books. Similarly a Science student picks 5 Science related books.

   Therefore there is no uniformity in the collection by different students. Hence "the collection of 5 good books in the school library" is not a well defined set.

   In the above example, it is not possible to decide whether a given book belongs or does not belong to the collection of objects.

   If the same example is considered as "The collection of new text books of X class", the collection is considered as a set.

Examples of sets :

   1) The collection of English alphabets

     a, b, c, d, ... x, y, z

   2) The collection of Natural numbers

     1, 2, 3, ...........

   3) The collection of Prime numbers

     2, 3, 5, 7,..............

   4) The collection of the devices of a computer. The input device, the central processing unit, the output device.

Examples of collections that are not sets

   1) The collection of 5 tasty sweet dishes.

   2) The collection of interesting novels.

   3) The collection of easy problems in X Maths text book.

   4) The collection of beautiful dresses.

* A Set is denoted by upper case letters A, B, C, D,.......... etc.

We have seen the different sets of numbers in the chapter Real Numbers.

Let us recall those sets.

   (i) N = {1, 2, 3, 4, ......}

   (ii) W = {0, 1, 2, 3, 4, ......}

   (iii) Z = {... -3, -2, -1, 0, 1, 2, 3, ......}

   (iv) Q = {p/ q, q ≠ 0, p , q ∊ Z}

   (v) Q' = 

  All these sets are well defined collections.

Meaning of the symbols '' and '':

    All the objects in the set are called elements of that set.

Example: Set of all the months in year that have 31 days.

A = {January, March, May, July, August, October, December}

We do not have all the twelve months in the set A as there are months that do not have 31 days in them.

1. Consider the month 'May'. It has 31 days. Hence we have written in the set A.

The month May is an element in the set A and we say that it belongs to set A and denoted using the symbol '∊'.

   May ∊ A

2. Consider the month 'June'. It is not written in the set A as it has only 30 days. We say that June does not belong to the set A and denoted using the symbol '∊'.

   June ∊ A

* Left hand side of the symbols  and  are elements and the right hand side of the symbols are sets.

Roster form of writing a set :

   When all the elements in the set are listed down separated by commas within curly brackets, the set is said be written in Roster form.

   All the above sets are written in Roster form.

* Consider the set B of all multiples of 2 less than 20

   B = {2, 4, 6, 8, .........., 18}

We can write the multiples of 2 in the following manner

   B = {18, 16, 14, 12, .........., 2} or

   B = {2, 12, 4, 14, 6, 16, 8, 18, 10}

     that is the order in which the elements in the set are listed is immaterial.

* Consider the following example

   The set M of all the letters in the word 'mathematics'

   M = {m, a, t, h, e, i, c, s}

While writing the set in the Roster form, the elements are not repeated i.e. the letters that are repeated in the word mathematics (m, a, t) are not written twice in the set.

From the above two examples we can say that when a set is written in the Roster form.

   (i) The order of listing the elements is immaterial

   (ii) The elements in the set are not repeated.

Set-builder form of writing sets

 Instead of listing the elements, if a set is defined in words with the common property of its elements, it is said to be written in the set builder form.

e.g.: B = {x/ x is a natural number less than 10}

      C = {x/ x is a day in the week}

      D = {x/ x is a colour in the rainbow}

  All the above sets starts with a symbol x followed by a vertical line then the common property of its elements. These sets are written in set builder form.

Types of sets

       Consider the following sets

       1) X = {x/ x is a vowel in the word sky}

       2) Y = {x/ x is a primary colour that begins with I}

       3) Z = {x/ x + 3 = 5 and x is a natural number greater than 2}

The letters of the word sky are s, k and y.

 All these are consonants and not vowels.

∴ The set X does not contain any elements.

In the second example, the primary colours are Red, Blue and Green. All these colours do not start with the letter 'I'. Therefore the set Y does not contain any element. It is empty.

In the third example, two conditions are to be satisfied i.e. x + 3 = 5 and x is a natural number greater than 2.

x + 3 = 5 ⇒ x = 2 but as the second condition says that the number must be greater than 2 we are left with no number that satisfies both the conditions.

∴  Set Z is also an empty set.

Empty Set

A set which does not contain any element is called an empty set or null set or a void set.

It is denoted by the symbol ∅ or { }.

Finite and Infinite Sets :

Consider the examples

1) A = { x/x is a function key on the key board of a computer}

2) B = {x/x is a letter in the word ASSESSMENT}

3) C = {x/x is a point on a straight line}

4) D = {2, 4, 6, 8, .....}

In the first two examples, the elements of the sets are limited i.e. finite.

     A = {F1, F2, F3, ..... F12}

     B = {A, S, E, M, N, T}

In the third and fourth examples, the elements are unlimited i.e. infinite.

  C = {p1, p2 , p3, ......}

  D = {2, 4, 6, .....}

* The sets with finite number of elements are called finite sets.

* The sets with infinite number of elements are called infinite sets.

* The number of elements in a set is called the cardinal number of the set.

In the the above examples

   n (A) = 12

   n(B) = 6

   n(C) is not defined

   n(D) is not defined

The cardinal number of a null set is zero. n(∅) = 0
 

Universal set and subsets :

In the first chapter, we have discussed about various sets of numbers.

Let us recall those sets

N = {1, 2, 3, 4, ...}

W = {0, 1, 2, 3, 4, ...}

Z = {... -3, -2, -1, 0, 1, 2, 3,...}

Q = {p/q, q ≠ 0, p, q ∈ Z}

  As we can see that all the natural numbers are present in the set of whole numbers and the set of whole numbers in integers and these in turn in rational numbers.

All these numbers along with the Irrational numbers are present in the set of Real Numbers.

The set of Real Numbers is the Universal Set. All the remaining sets are the subsets of Real Numbers.

The Universal set is denoted by μ and is usually represented by a rectangle.

All the subsets are represented by closed curves.

we observe that Q ⊂ R and Q' ⊂ R But Q ⊄ Q'

* When we say that A is not a subset of B we write A ⊄ B and it means that there is at least one element in A that is not an element in B.
 

EQUAL SETS

* If A ⊂ B and B ⊂ A then A = B. i.e., If A is the sub set of B and B is the sub set of A then A and B are equal sets.

e.g.: X = {x/x is a letter in the world smile}) then X = {S, M, I, L, E}

         Y = {x/x is a letter in the word MILES}

then Y = {M, I, L, E, S}

As the elements of X and Y are same X = Y

* Every set is a subset of itself.

* Empty set is the subset of every set.
 

To list all the subsets of a given set

        Let A = {p, q, r}

  To list the subsets of A let all the sets with single element be considered first, then the sets with two elements at a time from set A and finally all the three elements

i.e. A1 = ∅

       A2 = {p}

       A3 = [q}

       A4 = {r}

       A5 = {p, q}

       A6 = {p, r}

       A7 = {q, r}

       A8 = {p, q, r}

  There are 8 sub sets of the set A. In general if the set A has n elements then the number of subsets of A are 2n.
 

Venn diagrams :

      Venn-Euler diagram or simply Venn-diagram is a way of representing the relationships between sets.

1) Consider the following sets

     μ = {x/x is a letter in the word school}

     A = {x/x is a letter in the word cool}

     then μ = {s, c, h, o, l}

     and A = {c, o, l}

Let's represent these two sets using Venn diagram.

2) Consider one more set

     B = {x/x is a vowel in the word moon}, then
             

     Here B ⊂ A

3) Let μ = {1, 2, 3, ... 10}

     Set of primes less than 10

     P = {2, 3, 5, 7}

     Set of even numbers less than 10

     E = {2, 4, 6, 8}

4) Let μ = {1, 2, 3, 4,... 10}

     Set of even numbers less than 10

     E = {2, 4, 6, 8}

     Set of odd numbers less than 10

     O = {1, 3, 5, 7, 9}

There is no element in common between the set 'E' and set 'O'.

Such sets are called as disjoint sets.
 

Basic operations on sets :

Union of sets

  Suppose that three friends Anita, Bhanu & Chetan went to a shop to purchase stationary items. Anita bought a pencil and an eraser. Bhanu bought a pen, a marker and an eraser. They both kept these things in a cover. Chetan bought a chart and a thermocol sheet and carried these things in his hand. Let us represent the things bought by them using sets.

  A = {pencil, eraser}

  B = {pen, marker, eraser}

  C = {chart, thermocol sheet}

  What are the things they kept in the cover? The cover has the things bought either by Anita or by Bhanu.

  The set of things that are in the cover are {pencil, eraser, pen, marker}

  These things which are bought either by Anita or by Bhanu or by both is represented by the union of sets denoted by A ∪ B and is read as A Union B.

Thermocol sheet or the chart does not belong to A ∪ B.

A ∪ B = {x/x ∈ A or x ∈ B}

The Venn-diagram to represent A ∪ B is

Q: Let A = {2, 3, 5, 7, 11}, B = {2, 4, 6, 8} then represent A ∪ B by Venn diagram.

        A ∪ B = {2, 3, 4, 5, 6, 7, 8, 11}

Intersection of sets

   In the above stationary items example

   If we want to find the set of items that are bought both by Anita and Bhanu we get {eraser}. This set is denoted by A ∩ B and is read as 'A intersection B'.

   A ∩ B = {x/x ∈ A and x ∈ B}

Venn diagram to represent A ∩ B

Disjoint sets

    Let us again consider the items bought by Anita and Chetan this time.

    A = {pencil, eraser}

    C = {chart, thermocol sheet}

    Do they bought any thing in common?

    Nothing.

The sets are said to be disjoint sets.

* Two sets are said to be disjoint if they do not have any element in common.


 

Difference of sets

  The difference of sets A and B is the set of elements which belong to A but do not belong to B. We denote the difference of A and B by A-B and read as A minus B.

     A - B = {x/x ∈ A and x ∉ B}

     Similarly

     B - A = {x/x ∈ B and x ∉ A}

     A - B ≠ B - A.

Consider the Sets

     A = {2, 3, 5, 7}

     B = {2, 4, 6, 8}

     then A ∪ B = {2, 3, 4, 5, 6, 7, 8}

     A ∩ B = {2}

We observe that

     n(A) = 4, n(B) = 4, n (A ∪ B) = 7 and n(A ∩ B) = 1

     n(A) + n(B) - n(A ∩ B) = 4 + 4 - 1 = 7

 also n(A ∪ B) = 7

We observe that

     n(A ∪ B) = n(A) + n(B) - n (A ∩ B).
 

Examples:

1. Let A = {x / = - 3 x 2, x ∈ Z} B = {y / y is a factor of 12, y > 2} Write the sets A and B in roster form find  i) A ∪ B  ii) A ∩ B  iii) A - B iv) B - A from the venn Diagram. What do you infer?  (8 marks) 
Sol: We have, A = {x / - 3 x 2, x ∈ Z}
i.e., A = {-3, -2, -1, 0, 1, 2,}
also, B = {y / y is a factor of 12, y > 2}
B = {3, 4, 6, 12}
i) A ∪ B = { -3, -2, -1, 0, 1, 2, 3, 4, 6, 12}
ii) A ∩ B = { },
A and B are disjoint sets.
iii) A - B = {-3, -2, -1, 0, 1, 2} As A ∩ B = φ, A - B = A
iv) B - A = {3, 4, 6, 12} As A ∩ B = φ, B - A = B

 

2. Write all the subsets of the set A = {p, q, r} (4 Marks)
Sol:
A1 = { }
       A2 = {p}
       A3 = {q}
       A4 = {r}
      A5 = {p, q}
     A6 = {p, r}
     A7 = {q, r}
     A8 = {p, q, r} 

 

3. If A = {2, 3, 4, 5, 6}, B = {3, 6, 9, 12, 15} show that n(A ∪ B) = n(A) + n(B) − n(A ∩ B) (4 Marks)
Sol: Given, A = {2, 3, 4, 5, 6}
                ⇒ n(A) = 5 B = {3, 6, 9, 12, 15}
               ⇒ n(B) = 5 A ∪ B = {2, 3, 4, 5, 6} ∪ {3, 6, 9, 12, 15} = {2, 3, 4, 5, 6, 9, 12, 15}
               ⇒ n(A ∪ B) = 8 A ∩ B = {2, 3, 4, 5, 6} ∩ {3, 6, 9, 12, 15} = {3, 6}
               ⇒ n (A ∩ B) = 2
              n(A ∪ B) = 8 …...(1)
               n(A) + n(B) - n(A ∩ B) = 5 + 5 − 2 = 8 …. (2)
        From (1) & (2)
        n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

 

4. Draw venn diagrams to represent. 

5. Write an example for i) Finite Set ii) Infinite Set (2 Marks)
Sol: i) A = {a, e, i, o, u} is a finite set.
ii) B = {2, 4, 6, 8, 10, 12...} is an infinite set.

 

V. Padma Priya

Posted Date : 26-06-2021

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

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