Sets Class 11

📝 Sets

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●A given collection of objects is said to be well-defined if we can definitely say whether a given particular object belongs to the collection or not. 

Set 

➤ A well-defined collection of objects is called a set. 

● The objects in a set are called its members or elements. We denote sets by capital letters `A, B, C, X, Y, Z, etc.`

● If `a` is an element of a set `A`, we write `a in A`, which means that `a` belongs to `A` or `a` is an element of `A`.

● If `a` does not belong to `A`, then we write, `a notin A` 

 The symbols for the special sets

`N:`  the set of all-natural numbers 

`Z:`  the set of all integers 

`Q:`  the set of all rational numbers 

`R:`  the set of real numbers 

`Z ^+:`  the set of positive integers 

`Q^+:`  the set of positive rational numbers, and 

`R^+:`  the set of positive real numbers.

There are two methods of representing a set :

1. Roster or tabular form
2. Set-builder form. 

(1) Roster or tabular form

In roster form, all the elements of a set are listed, and the elements are 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}.` 

👉 In roster form, the order in which the elements are listed is immaterial.

👉 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.

 

2. 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 `V = {x : x` is a vowel in English alphabet`}` 

👉 The above description of the set V is read as “the set of all `x` such that `x` is a vowel of the English alphabet”. In this description the braces stand for “the set of all”, the colon stands for “such that”. 

The Empty Set

Definition  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 `phi` or `{ }.` 

We give below a few examples of empty sets. 

(i) Let `A = {x : 1 < x < 2, x` is a natural number`}`. Then `A` is the empty set, because there is no natural number between 1 and 2. 

(ii) `B = {x : x ^2 – 2 = 0` and `x` is rational number`}`. Then `B` is the empty set because the equation `x ^2 – 2 = 0` is not satisfied by any rational value of x. 

(iii) `C = {x : x` is an even prime number greater than 2`}`.Then C is the empty set, because 2 is the only even prime number. 

`(iv) D = { x : x ^2 = 4`, x is odd `}`. Then D is the empty set, because the equation `x^ 2 = 4` is not satisfied by any odd value of x.

Finite and Infinite Set

Definition  A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Consider some examples : 

(i) Let W be the set of the days of the week. Then W is finite. 

(ii) Let S be the set of solutions of the equation `x ^2 –16 = 0`. Then S is finite. 

(iii) Let G be the set of points on a line. Then G is infinite.

Equal Set

Definition  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`. 

We consider the following examples : 

(i) Let `A = {1, 2, 3, 4}` and `B = {3, 1, 4, 2}`. Then `A = B`. 

(ii) 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.

Subsets

Definition  A set `A` is said to be a subset of a set `B` if every element of `A` is also an element of `B`. 

`A ⊂ B`  if `a ∈ A ⇒ a ∈ 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 `A ⊄ B.`

👉  `A ⊂ B and B ⊂ A ⇔ A = B`

👉  Every set `A` is a subset of itself, i.e., `A ⊂ A.`

👉  `phi` is a subset of every set.

Let `A` and `B` be two sets. If `A ⊂ B` and `A ≠ B` , then `A` is called a proper subset of `B` and `B` is called superset of `A`. 

For example, `A = {1, 2, 3}` is a proper subset of `B = {1, 2, 3, 4}.`

Practical Problems on Union and Intersection of Two Sets

If `A` and `B` are finite sets such that `A ∩ B = phi`, 

then  `n (A ∪ B) = n (A) + n (B)`.

 If `A ∩ B ≠ phi`,  then  `n (A ∪ B) = n (A) + n (B) – n (A ∩ B)`

Work in progress.

References 

1. NCERT Mathematics Class 11
2. https://en.wikipedia.org/wiki/Georg_Cantor