Set Theory # 1

Set: Collection of Well Defined Objects

What do we mean by Well defined?

Which of these is a set?

a) “All Beautiful Girls in Kolkata”

b) All Tall Boys in class XI

c) All cricketers who have scored more than

10000 runs in ODI’s till year 2010

d) {A particular atom which took part in the

collision in a nuclear reaction, The 3rd

largest satellite of Jupiter, the fan in the

right corner of Lok Sabha on 12th January

1994}

e) The tallest boy in class IX at EDUDIGM

Sunday Batch

f) Set of all Miss Kolkata’s till the year 2010

Figure out yourself: √2 is irrational

Roster Form

Set Builder Form

Venn-Diagram

Elements

Set

Universal Set

Intervals (Write in Set Builder form and plot on the Real

Line)

(Closed Interval) ,1 2-

(Open Interval) (1 2)

,1 2)

(1 2-

Empty Sets * +

Universal Sets

Complement

Comments:

*Godel’s Incompleteness Theorem

Union and Intersection

Find and plot the following on the number line.

a) (1 2) (1 5 2 5)

b) ,1 2- ,1 5 2 5- ,1 25 1 75-

Shade

Some Basic Results

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

Difference of Sets

Shade

Complement

( )

( )

Prove 2nd

Result

Cardinality

Which has more elements?

The set of Natural Numbers or whole numbers?

The set of Integers or the set of Natural

Numbers?

Sub Set of a Set

Power Set

Cardinality is:

Equal Sets

Cartesian Product of Sets

*1 2+ * +

Cardinality of Cartesian Product is:

A B A B

A B

A B

A B A B

Important Results

If A, B and C are finite sets and U be the finite

universal set, then

(1) ( ) ( ) ( )– ( )

(2) ( – ) ( )– ( )

(3) ( ) ( ) ( ) ( )

– ( )– ( )– ( ) (

)

Prove the Above results Using Venn Diagrams

Q) Let A = {1, 2, 3, 4, 5}; B = {2, 3, 6, 7}. Then the

number of elements in (A × B) (B × A) is

Q) In a class of 55 students, the number of

students studying different subjects are 23 in

Mathematics, 24 in Physics, 19 in Chemistry, 12

in Mathematics and Physics, 9 in Mathematics

and Chemistry, 7 in Physics and Chemistry and

4 in all the three

subjects. The

number of students

who have taken

exactly one subject is

𝑛𝑎 𝑛𝑏

Problem Solving…

1) There are 100 numbers written in a line

with the property that the sum of the first k

numbers is . Find the 10th number.

2) Find the 10th term of a series whose nth term

is 3 2

3) Explain this seemingly strange but striking

result: 1 1

1 3 2

1 3 5 3

1 3 5 7 4

1 3 5 7 9 5 …

4) Find the sum of all terms of a chess board if

each square ( ) is numbered as

a) b) c)

5) Which is the smallest Positive real number?

6) Explain this fallacy here:

1 1 1 1 1 1…

1 (1 1 1 1 1 1… )

1 1 2

S is the sum of integers. How can it be half?

7) Which of the following is true?

a) 1 1 b) 1 1 c) 1 2

d) 1 2 e) 1 1 f) 1 2

8) 9̅ 1?

9) Evaluate

√2 √2 √2…

10) Find .1

/ .1

/ .1

/ .1

/…

11) Find the roots of 3 2 3 1

0

12) Show that 16 17 18 19 is multiple

of 70

13) Factorize 1

14) Simplify ( )( )( )(

)… ( )

15) Find 1 2 3 4 5 6 …

99 100

16) A cube is cut into 27 equal cubes. All the

faces of the cube were initially red. What is

the number of faces with exactly two red

coloured surfaces? How would the answer

change if there were 125(5x5x5) sub cubes

formed?

17) If ( ) ( ) ( ) for all values of x

and y, and (1) 1, Find (100)

18) Explain this seemingly strange but striking

result:

1 1 1 3 2 1 3 5

3 1 3 5 7

4 1 3 5 7 9 5 …

19) By only interchanging Rows/ Columns can

you convert a chess board to the 2 here?

20) Can there be a perfect square among these

numbers? 1 11 111 1111 11111 … (Other

than 1?)

21) Look at odd perfect squares… They all leave

remainder 1 on division by 8. Why?

22) Express the following as the difference of

two perfect squares: 4

23) What is the number of times that 4 appears

if you were to write all the numbers from 0

to 999 one after another?

24) Evaluate

…

if there are 100 2’s

25) Prove:

…