## TABULAR FORM

Elementary Mathematics**A well defined collection of {distinct}objects is called a set.**

- The objects are called the elements or members of the set.
- Sets are denoted by capital letters A, B, C …, X, Y, Z.
- The elements of a set are represented by lower case letters a, b, c, … , x, y, z.
- If an object x is a member of a set A we write x ∈A, which reads “x belongs to A” or “x is in A” or “x is an element of A”, otherwise we write x ∉A, which reads “x does not belong to A” or “x is not in A” or “x is not an element of A”.

**TABULAR FORM**

Listing all the elements of a set, separated by commas and enclosed within braces or curly brackets{}.

**EXAMPLES**

In the following examples we write the sets in Tabular Form. A = {1, 2, 3, 4, 5} is the set of first five Natural Numbers. B = {2, 4, 6, 8, …, 50} is the set of Even numbers up to 50. C = {1, 3, 5, 7, 9 …} is the set of positive odd numbers.

*NOTE *

The symbol “…” is called an ellipsis. It is a short for “and so forth.”

*DESCRIPTIVE FORM: *

Stating in words the elements of a set.

**EXAMPLES **

Now we will write the same examples which we write in Tabular

Form ,in the Descriptive Form. A = set of first five Natural Numbers.( is the Descriptive Form ) B = set of positive even integers less or equal to fifty.

( is the Descriptive Form )

C = {1, 3, 5, 7, 9, …} ( is the Descriptive Form )

C = set of positive odd integers. ( is the Descriptive Form )

SET BUILDER FORM:

Writing in symbolic form the common characteristics shared by all the elements of the set.

**EXAMPLES: **

Now we will write the same examples which we write in Tabular as well as Descriptive Form

,in Set Builder Form . A = {x ÎN / x<=5} ( is the Set Builder Form) B = {x Î E / 0 < x <=50} ( is the Set Builder Form) C = {x ÎO / 0 < x } ( is the Set Builder Form)

**SETS OF NUMBERS: **

**1. Set of Natural Numbers **

N = {1, 2, 3, … }

**2. Set of Whole Numbers **

W = {0, 1, 2, 3, … }

**3. Set of Integers **

Z = {…, -3, -2, -1, 0, +1, +2, +3, …} = {0, ±1, ±2, ±3, …} {“Z” stands for the first letter of the German word for integer: Zahlen.}

**4. Set of Even Integers **

E = {0, ± 2, ± 4, ± 6, …}

**5. Set of Odd Integers **

O={± 1, ± 3, ± 5, …}

**6. Set of Prime Numbers **

P = {2, 3, 5, 7, 11, 13, 17, 19, …}

**7. Set of Rational Numbers (or Quotient of Integers) **

Q = {x | x = ; p, q ∈Z, q ≠ 0}

**8. Set of Irrational Numbers **

Q=Q′ = { x | x is not rational} For example, √2, √3, π, e, etc.

**9. Set of Real Numbers **

R=Q ∪ Q′

**10. Set of Complex Numbers **

C = {z | z = x + *i*y; x, y ∈ R}

**SUBSET: **

If A & B are two sets, A is called a subset of B, written A ⊆ B, if, and only if, any element of A is also an element of B. Symbolically: A ⊆ B ⇔ if x ∈ A then x ∈ B

**REMARK: **

1. When A ⊆ B, then B is called a superset of A.

2. When A is not subset of B, then there exist at least one x ∈ A such that x ∉B.

3. Every set is a subset of itself.

**EXAMPLES:**

Let

A = {1, 3, 5} B = {1, 2, 3, 4, 5}

C = {1, 2, 3, 4} D = {3, 1, 5}

Then

A ⊆ B ( Because every element of A is in B )

C ⊆ B ( Because every element of C is also an element of B )

A ⊆ D ( Because every element of A is also an element of D and also note

that every element of D is in A so D ⊆ A )

and A is not subset of C .

( Because there is an element 5 of A which is not in C ) ** **

**EXAMPLE: **

The set of integers “Z” is a subset of the set of Rational Number “Q”, since every integer ‘n’ could be written as:

n = ^{n }∈Q

1

Hence Z ⊆ Q.

PROPER SUBSET

Let A and B be sets. A is a proper subset of B, if, and only if, every element of A is in B but there is at least one element of B that is not in A, and is denoted as A ⊂ B.

EXAMPLE:

Let A = {1, 3, 5} B = {1, 2, 3, 5} then A ⊂ B ( Because there is an element 2 of B which is not in A).

**EQUAL SETS: **

Two sets A and B are equal if, and only if, every element of A is in B and every element of B is in A and is denoted A = B. Symbolically: A = B iff A ⊆ B and B ⊆ A

**EXAMPLE: **

Let A = {1, 2, 3, 6} B = the set of positive divisors of 6 C = {3, 1, 6, 2} D = {1, 2, 2, 3, 6, 6, 6} Then A, B, C, and D are all equal sets.

**NULL SET: **A set which contains no element is called a null set, or an empty set or a void set. It is denoted by the Greek letter ∅ (phi) or { }.

EXAMPLE

A = {x | x is a person taller than 10 feet} = ∅ ( Because there does not exist any human being which is taller then 10 feet ) B = {x | x2 = 4, x is odd} = ∅ (Because we know that there does not exist any odd whose square is 4)

**REMARK **

∅ is regarded as a subset of every set.

**EXERCISE: **

Determine whether each of the following statements is true or false.

**a.**x ∈ {x} TRUE ( Because x is the member of the singleton set { x } )

a.{x}⊆ {x} TRUE ( Because Every set is the subset of itself. Note that every Set has necessarily tow subsets ∅ and the Set itself, these two subset are known as Improper subsets and any other subset is called Proper Subset)

**a.**{x} ∈{x} FALSE ( Because { x} is not the member of {x} ) Similarly other

d. | {x} ∈{{x}} | TRUE |

e. | ∅ ⊆ {x} | TRUE |

f. | ∅ ∈ {x} | FALSE |

**UNIVERSAL SET: **

The set of all elements under consideration is called the Universal Set. The Universal Set is usually denoted by U.

**VENN DIAGRAM: **

A Venn diagram is a graphical representation of sets by regions in the plane. The Universal Set is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle.

U

**FINITE AND INFINITE SETS:** A set S is said to be finite if it contains exactly *m* distinct elements where *m *denotes some non negative integer.

In such case we write |S| = *m* or n(S) = *m *A set is said to be infinite if it is not finite.

**EXAMPLES: **

1. The set S of letters of English alphabets is finite and |S| = 26

2. The null set ∅ has no elements, is finite and |∅| = 0

3. The set of positive integers {1, 2, 3,…} is infinite.

**EXERCISE: **

Determine which of the following sets are finite/infinite.

1. A = {month in the year} FINITE

2. B = {even integers} INFINITE

3. C = {positive integers less than 1} FINITE

4. D = {animals living on the earth} FINITE

5. E = {lines parallel to x-axis} INFINITE

6. F = {x ∈R | x^{100} + 29x^{50} – 1 = 0} FINITE

7. G = {circles through origin} INFINITE MEMBERSHIP TABLE:

A table displaying the membership of elements in sets. To indicate that an element is in a set, a 1 is used; to indicate that an element is not in a set, a 0 is used.

Membership tables can be used to prove set identities.

A |
Ac |

1 |
0 |

0 |
1 |

The above table is the Member ship table for Complement of A. now in the above table note that if an element is the member of A then it can`t be the member of A^{c} thus where in the table we have 1 for A in that row we have 0 in A^{c}.

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