1 set theory class handout modified
TRANSCRIPT
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:
…