Set of Data Points (p Nitin)

All branches of Math can be brought into the frame work of set theory. Let us recall what we have learnt the sets by answering the following questions: Define a set, empty set, disjoint sets, subsets, power set, universal set, complement of a set, operations on sets and give some examples.

German mathematician Gottlob presented the set theory as principles of logic. Bertrand Russell (1872-1970 A.D) and Paul Hal mos proposed paradoxes in set theory. Cantor is the father of modern theory of sets.The set of data points example problems and practice problems are given below.

Example problems for learning sets of data points :

Learning set of data points-Example problem 1:

In a higher secondary class, 66 play football, 56 play hockey, 63 play cricket, 27 play both football and hockey, 25 play hockey and cricket, 23 play cricket and football and 5 do not play any game. If the strength of the class is 130.

Calculate (i) the number who play only two games,

(ii) the number who play only football and

(iii) number of students who play all the three games

Solution:

Let F, H, C be the sets those who play football, hockey and cricket respectively.
Strength of the class n(?) = 130
No. of students who play football = n(F) = 66
No. of students who play hockey = n(H) = 56
No. of students who play cricket = n(C) = 63
No. of students who play football and hockey = n(FnH) = 27

No. of students who play hockey and cricket = n(H n C) = 25
No. of students who play cricket and football = n(C n F) = 23
Let the no. of students who play all the three games = n(F n H n C) = x
No. of students who do not play any game = 5

No. of students who play only football
= 66 - (27 - x + x + 23 - x)

= 66 - (50 - x) = 66 - 50 + x = 16 + x
No. of students who play only hockey
= 56 - (27 - x + x + 25 - x)
= 56 - (52 - x) = 4 + x
No. of students who play only cricket
= 63 - (23 - x + x + 25 - x) = 63 - (48 - x) = 15 + x

No. of students who play at least one game is
n(F?H?C) = n(?) - No. of students who do not play = 130 - 5 = 125
From the diagram, n (F ? H ? C) = 16 + x + 27 - x + 4 + x + 23 - x + x + 25 - x + 15 + x = 110 + x
110 + x = 125, x = 125 - 110 = 15.
i) No.of students who play only two games = 27 - x + 23 - x + 25 - x = 75 - 3x
= 75 - 3(15) = 75 - 45 = 30
ii) No. of students who play only football = 16 + x = 16 + 15 = 31
iii) No. of students who play all the three games = x = 15

Learning set of data points - Proof1

1. Union of two sets is commutative

A ? B = B ? A.

For example, given A = {-2, 3, -15, 7} and B = {3, 5, 9, 11} we see that
A ? B = {-15, -2, 3, 5, 7, 9, 11}, B ? A = {-15, -2, 3, 5, 7, 9, 11} ? A ? B = B ? A

Learning set of data points- Proof 2:

2. Intersection of two sets is commutative

A n B = B n A.

For example, given A = {7, 5, 2, ,6} and B = {3, 6, 7, 12}.

We see that A n B = {7, 6}; B n A = {7, 6} ? A n B = B n A.

Practice problem for learning set of data points :

Practice problem:

Verify the commutative laws of union and intersection of the following sets
(i) A = {1, 2, 3, 4, 5, 6}, B = {5, 6, 7, 8} (ii) A = {a, e, i, o, u}, B = {a, u}

Ans: (1) AnB = { 5, 6}, AUB = {1, 2, 3, 4, 5, 6, 7, 8}

(2) AnB = {a, u}, AUB = {a, e, i, o, u}

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