SETS-Notes
Maths - Notes
SETS
A set is a well-defined collection of distinct objects, called elements or members of the set.
The phrase well-defined means that it must be possible to clearly decide whether a particular object belongs to the set or not.
Examples
- The collection of natural numbers less than 5 is a set: {1,2,3,4}.
- The collection of vowels in the English alphabet is a set.
- The collection of “beautiful flowers” is not a set, since the word beautiful is subjective.
Below a few more examples of sets used particularly in mathematics, viz.
- \(\mathbb{N}\) : the set of all natural numbers
- \(\mathbb{Z}\) : the set of all integers
- \(\mathbb{Q}\) : the set of all rational numbers
- \(\mathbb{R}\) : the set of real numbers
- \(\mathbb{Z^+}\) : the set of positive integers
- \(\mathbb{Q^+}\) : the set of positive rational numbers, and
- \(\mathbb{R^+}\) : the set of positive real numbers.
Methods of representing a set :
There are two methods of representing a set :
- Roster or tabular form
- Set-builder form
Roster or tabular form
In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example, the set of all even positive integers less than 7 is described in roster form as {2, 4, 6}. Some more examples of representing a set in roster form are given below :
- The set of all natural numbers which divide 42 is {1, 2, 3, 6, 7, 14, 21, 42}
- The set of all vowels in the English alphabet is {a, e, i, o, u}
- The set of odd natural numbers is represented by {1, 3, 5, . . .}. The dots tell us that the list of odd numbers continue indefinitely.
Set-builder form
In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V, we write \[\mathrm{V = \{x : x \text{ is a vowel in English alphabet}}\}\] Some more examples of representing a set in Set-builder form are given below :
- A= {x : x is a natural number which divides 42}
- B= {y : y is a vowel in the English alphabet}
- C= {z : z is an odd natural number}
Important Points
- In roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as {1, 3, 7, 21, 2, 6, 14, 42}.
- It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct. For example, the set of letters forming the word ‘SCHOOL’ is { S, C, H, O, L} or {H, O, L, C, S}. Here, the order of listing elements has no relevance.
The Empty Set
A set which does not contain any element is called the empty set or the null set or the void set.
According to this definition, B is an empty set while A is not an empty set. The empty set is denoted by the symbol \(\mathrm{\phi}\) or { }.
Below a few examples of empty sets.
- Let \[\mathrm{A = \{x : 1 < x < 2, x\text{ is a natural number}\}}\] Then A is the empty set, because there is no natural number between 1 and 2.
- \[\mathrm{B = \{x : x^2 – 2 = 0\text{ and }x\text{ is rational number}\}}\] Then B is the empty set because the equation \(\mathrm{x^2 – 2 = 0}\) is not satisfied by any rational value of x.
- \[\mathrm{C = \{x : x\text{ is an even prime number greater than 2}\}}\] Then C is the empty set, because 2 is the only even prime number.
- \[\mathrm{D = \{ x : x^2 = 4,\, x\text{ is odd} \}}\] Then D is the empty set, because the equation \(\mathrm{x^2 = 4}\) is not satisfied by any odd value of x.
Finite and Infinite Sets
A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.
Examples
- Let W be the set of the days of the week. Then W is finite.
- Let S be the set of solutions of the equation \(\mathrm{x^2 –16 = 0}\). Then S is finite.
- Let G be the set of points on a line. Then G is infinite.
Important Note
- All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.
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 ≠ B.
Examples
- Let \[\mathrm{A = \{1, 2, 3, 4\}}\] and \[\mathrm{B = \{3, 1, 4, 2\}}\] Then \[\mathrm{A = B}\]
- Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.
Important Note
- A set does not change if one or more elements of the set are repeated. For example, the sets \[\mathrm{A = \{1, 2, 3\}}\] and \[\mathrm{B = \{2, 2, 1, 3, 3\}}\] are equal, since each element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set.
Subsets
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 other words, \(\mathrm{A \subset B}\) if whenever \(\mathrm{a \in A}\), then \(\mathrm{a \in B}\). It is
often convenient to
use the symbol “\(\mathrm{\Rightarrow}\)” which means implies. Using this symbol, we can write the definiton
of subset as follows:
\[\mathrm{A \subset B \text{ if } a \in A \Rightarrow a \in B}\]
We read the above statement as “A is a subset of B if a is an element of A
implies that a is also an element of B”
If A is not a subset of B, we write \(\mathrm{A \not\subset B}\).
Examples
- The set \(\mathbb{Q}\) of rational numbers is a subset of the set \(\mathbb{R}\) of real numbes, and we write \[\mathrm{\mathbb{Q} \subset \mathbb{R}}\]
- If A is the set of all divisors of 56 and B the set of all prime divisors of 56, then B is a subset of A and we write \[\mathrm{B \subset A}\]
- Let \[\mathrm{A = \{1, 3, 5\}}\] and \[\mathrm{B = \{x : x\, \text{ is an natural number less than } 6\}}\] Then \[\mathrm{A \subset B \text{ and } B \subset A}\] and hence \[\mathrm{A = B}\]
- Let \[\mathrm{A = \{ a, e, i, o, u\}}\] and \[\mathrm{B = \{ a, b, c, d\}}\] Then A is not a subset of B, also B is not a subset of A.
Universal Set
A universal set is a set that contains all the elements under consideration in a particular context or discussion.
It is usually denoted by the symbol \(\mathrm{U}\).
The universal set depends entirely on the situation being studied and therefore may change from one problem to another.
Venn Diagrams
Most of the relationships between sets can be represented by means of diagrams which are known
as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883).
These diagrams consist of rectangles and closed curves usually circles. The universal set is represented
usually by a rectangle and its subsets by circles.
Union of sets
Let \(A\) and \(B\) be any two sets. The union of A and B is the set which consists of all the elements of \(A\) and all the elements of \(B\), the common elements being taken only once. The symbol \(\cup\) is used to denote the union. Symbolically, we write \(A \cup B\) and usually read as ‘\(A\) union \(B\)’
The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write. \[\mathrm{A \cup B = \{ x : x \in A \text{ or } x \in B \}}\] The union of two sets can be represented by a Venn diagram as shown in Fig 1.4.
Properties of the Operation of Union
- Commutative law \[\mathrm{A\cup B=B\cup A}\]
- Associative law \[\mathrm{(A\cup B)\cup C=A\cup (B\cup C)}\]
- Law of identity element, \(\phi\) is the identity of ∪ \[\mathrm{A\cup \phi=A}\]
- Idempotent law \[\mathrm{A\cup A=A}\]
- Law of U \[\mathrm{U\cup A=U}\]
Intersection of sets
The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write \[\mathrm{A \cap B = \{x : x \in A \text{ and } x \in B\}}\]
The shaded portion in Fig 1.5 indicates the intersection of A and B.
Properties of Operation of Intersection
- Commutative law \[\mathrm{A\cap B=B\cap A}\]
- Associative law \[\mathrm{(A\cap B)\cap C=A\cap (B\cap C)}\]
- Law of identity element, \(\phi\) is the identity of ∪ \[\mathrm{\phi \cap A = \phi}\] \[\mathrm{U \cap A = A}\]
- Idempotent law \[\mathrm{A\cap A=A}\]
- Distributive law \[\mathrm{A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )}\]
Difference of sets
The difference of the sets A and B in this order is the set of elements which belong to A but not to B. Symbolically, we write A – B and read as “A minus B”
Complement of a Set
Let \(U\) be the universal set and \(A\) a subset of \(U\). Then the complement of \(A\) is the set of all elements of \(U\) which are not the elements of \(A\). Symbolically, we write \(A^′\) to denote the complement of A with respect to \(U\). Thus, \[\mathrm{A′ = \{x : x \in U \text{ and } x \not\in A \}}\] Obviously \[\mathrm{A^′ = U – A}\]
We note that the complement of a set \(A\) can be looked upon, alternatively, as the difference between a universal set \(U\) and the set \(A\).
Properties of Operation of Intersection
- Complement laws \[\mathrm{A \cup A^′ = U}\] \[\mathrm{ A \cap A^′ = φ}\]
- De Morgan’s law \[\mathrm{(A \cup B)^′ = A^′ \cap B^′}\] \[\mathrm{(A \cap B)^′= A^′ \cup B^′}\]
- Law of double complementation \[\mathrm{(A^′)^′ = A}\]
- Laws of empty set and universal set \[\mathrm{ \phi^′ = U }\] and \[\mathrm{U^′ = \phi}\]
Summary
- A set is a well-defined collection of objects.
- A set which does not contain any element is called empty set.
- A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.
- Two sets A and B are said to be equal if they have exactly the same elements.
- A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of \(\mathbb{R}\).
- The union of two sets A and B is the set of all those elements which are either in A or in B.
- The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.
- The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.
- For any two sets A and B, \[\mathrm{(A \cup B)^′ = A^′ \cap B^′}\] and \[\mathrm{( A \cap B )^′ = A^′ \cup B^′}\]
Example-1
Write the solution set of the equation \(\mathrm{x^2 + x – 2 = 0}\) in roster form.
Solution
$$\begin{aligned}x^{2}+x-2&=0\\ x^{2}+2x-x-2&=0\\ x\left( x+2\right) -1\left( x+2\right) &=0\\ \left( x+2\right) \left( x-1\right) &=0\\ \Rightarrow x&=-2\\ x&=1 \end{aligned}$$Therefore solution set of given equation can be written roster form as \(\{-2, 1\}\)
Example-2
Write the set \(\mathrm{\{x : x \text{ is a positive integer and } x^2 < 40\}}\) in the roster form.
Solution
$$\begin{aligned}x^{2} \lt 40\\ x\Rightarrow \left\{ 1,2,3,4,5,6\right\} \end{aligned}$$The required numbers in a roster form is \(\mathrm{\{1,2, 3,4, 5,6\}}\)
Example-3
Write the set \(\mathrm{A = \{1, 4, 9, 16, 25, . . . \}}\)in set-builder form.
Solution
\[\mathrm{A = \{1,4, 9,16, 25..\}}\] Set in a set builder form $$A=\{ x:x= n^{2}, \text{ where } n\in \mathbb{N} \} $$Example-4
Write the set \(\left\{ \dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{5}{6},\dfrac{6}{7}\right\}\) in the set-builder form.
Solution
$$\begin{aligned} \left\{ x:x=\dfrac{n}{n+1},\; n\in \mathbb{N} \text{ and } 1\leq n\leq 6\right\} \end{aligned}$$Example-5
Find the pairs of equal sets, if any, give reasons:
\(\mathrm{A = \{0\}}\),
\(\mathrm{B = \{x : x > 15 and x < 5\}}\),
\(\mathrm{C = \{x : x – 5 = 0 \}}\),
\(\mathrm{ D=\{x: x^2=25\}}\),
\(\mathrm{E=\{x : x\text{ is an integral positive root of the equation } x^2 – 2x
–15=0\}}\).
Solution
$$\begin{aligned}A&=\left\{ 0\right\} \\\\ B&=\left\{ x:x >15,x \lt 5\right\} \\ &\Rightarrow \left\{ \ldots 0,1,2,3,4,16,17,18...\right\} \\\\ C&=\left\{ x:x-5=0\right\} \\ &\Rightarrow \left\{ 5\right\} \\\\ D&=\left\{ x:x^{2}=25\right\} \\ &\Rightarrow \left\{ -5,5\right\} \\\\ E&=\left\{ x:x^{2}-2x-15=0\right\} \\ &\Rightarrow x^{2}-2x-15=0\\ &\Rightarrow x^{2}-5x+3x-15=0\\ &\Rightarrow x\left( x-5\right) +3\left(\Rightarrow x-5\right)=0\\ &\Rightarrow\left( x-5\right) \left( x+3\right)=0\\ &\scriptsize \text{Solving for positive } x\\\\ E&=\left\{ 5\right\} \\\\ C=&E=\left\{ 5\right\} \end{aligned}$$Example-6
Consider the sets
\(\phi\), A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets:
(i) \(\phi\) . . . B
(ii) A . . . B
(iii) A . . . C
(iv) B . . . C
Solution
\(\phi\) \(\mathrm{A =\{1,\,3\}}\) \(\mathrm{B = \{1,\, 5,\,9\}}\) \(\mathrm{C = \{1,\, 3,\, 5,\,7,\, 9\}}\) $$\begin{aligned}(i)\quad\Phi &\subset B\\ (ii)\quad A&\not \subset B\\ (iii)\quad A&\subset C\\ (iv)\quad B&\subset C\end{aligned}$$Example-7
Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.
Solution
$$\begin{aligned}A&=\left\{ 2,4,6,8\right\} \\ B&=\left\{ 6,8,10,12\right\} \\\\ A\cup B&=\left\{ 2,4,6,8,10,12\right\} \end{aligned}$$Example-8
Consider the sets A and B of Example 7. Find A ∩ B.
Solution
$$\begin{aligned}A&=\left\{ 2,4,6,8\right\} \\ B&=\left\{ 6,8,10,12\right\} \\\\ A\cap B&=\left\{ 6,8\right\} \end{aligned}$$Example-9
Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution
$$\begin{aligned}A&=\left\{ 1,3,4,5,6\right\} \\ B&=\left\{ 2,4,6,8\right\} \\\\ A-B&=\left\{ 1,3,5\right\} \\\\ B-A&=\left\{ 8\right\} \\\\ A-B&\neq B-A\end{aligned}$$Example-10
Let V = { a, e, i, o, u } and B = { a, i, k, u}. Find V – B and B – V
Solution
$$\begin{aligned}V&=\left\{ a,e,i,0,u\right\} \\ B&=\left\{ a,i,k,u\right\} \\\\ V-B&=\left\{ e,0\right\} \\\\ B-V&=\left\{ k\right\} \\\\ V-B&\neq B-V\end{aligned}$$Example-11
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.
