venn diagram sets

About venn diagram 

The diagrams drawn to represent sets are called Venn diagrams or Eule -Venn diagrams. Here we represent the universal set U by points within rectangle and the subset A of the set U represented by the interior of a circle. If a set A is a subset of a set B then the circle representing A is drawn inside the circle representing B. If A and B are no equal but they have some common elements, then to represent A and B by two intersecting circles.Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah. 

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Let’s us consider a n example A class has 175 students. The following table shows the number of students studying one or more of the following subjects in this case

Subjects               No. of students

Mathematics             100

Physics                        70

Chemistry                   46

Mathematics and Physics                          30

Mathematics and Chemistry                     28

Physics and Chemistry                             23

Mathematics, Physics and Chemistry    18

How many students are enrolled in Mathematics alone, Physics alone and Chemistry alone? Are there students who have not offered any one of these subjects?

Detail Explanation of above problem venn diagram Question

Let P, C, M denote the sets of students studying Physics, Chemistry and Mathematics respectively.

Let a, b, c, d, e, f, g denote the number of elements (students) contained in the bounded region as shown in the diagram then

after solving we get g = 18, f = 10, e = 5, d = 12, a = 35, b = 13 and c = 60

∴ a + b + c + d + e + f + g = 153

So, the number of students who have not offered any of these three subjects

= 175 –153 = 22

 Number of students studying Mathematics only, c = 60

Number of students studying Physics only, a = 35

Number of students studying Chemistry only, b = 13.

CARTESIAN PRODUCT OF SETS 

The Cartesian product (also known as the cross product) of two sets A and B, denoted by A×B (in the same order) is the set of all ordered pairs (x, y) such that x∈A and y∈B. What we mean by ordered pair is that the pair(a, b) is not the same the pair as (b, a) unless a = b. It implies that A×B ≠ B×A in general. Also if A contains m elements and B contains n elements then A×B contains m×n elements.

Similarly we can define A×A = {(x, y); x∈A and y∈A}. We can also define cartesian product of more than two sets.

 e.g.  A₁× A₂×A₃ × . . . .× A_n = {(a₁, a₂, . . . , a_n): a₁ ∈A₁, a₂ ∈ A₂, . . . , a_n ∈ A_n}