Discrete mathematics

            It's about Discrete mathematics in this book  #Anthony Croft, Engineering Mathematices, 5e (2017) 

So i will choose exercise 5.2 #7 #8 #9  to slow everyone:

7. The sets A, B and C are given by A = {1, 3, 5, 7, 9},
B = {0, 2, 4, 6} and C = {1, 5, 9} and the universal

set, E = {0,1, 2, . . . ,9}.
(a) Represent the sets on a Venn diagram.
(b) State A ∪ B.
(c) State B ∩ C.
(d) State E ∩ C.
(e) State not A.
(f) State not B ∩not C.
(g) State not (B ∪C).

• Ans : 

a) Represent the sets on a Venn diagram :

The set containing all the numbers of interest is called the
universal set, E. E is represented by the rectangular region. Sets A , B and C are represented by the interiors of the circles and it is evident that 1,  3,  5,  7 and 9 are members of A while 0,
2,  4,  and 6 are members of B that 1,  5  and  9 are members of C.  The elements 1 ,  5 and 9 are common to both sets.

b) State A ∪ B :

The set which contains all the elements of A and those of B is

called the union of A and B, written as

                 b)   {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 9}


c) State B ∩ C :

the set B ∩ C has no elements we say the sets B and C are disjoint and write B ∩ C = empty set.

d) State E ∩ C :

The set which contains the elements common to both E and  C is called the intersection of E and C, written as

                   d)   {1 ,  5  ,  9}

e) State not A :

The set of members of E that are not in A is called the complement of A : 

                  e)     {0 , 2 , 4 , 6 , 8}

f) State not B ∩  not C : 

The set of members of E that are not in B  and not in C is called the complement of B and C : 

                   f)      {3 , 7 , 8}

g) State not (B ∪C) :

The set which contains all the elements of not A and those of not B is called the union of not A and not B, written as

                   g)      {3 , 7 . 8}

8.Use Venn diagrams to illustrate the following for
general sets C and D:

(a) C ∩D              (b) C ∪ D                (c) C ∩ not_D
(d) not (C ∪D)            (e) not (C ∩D)


Ans :

a) C ∩D :

The set which contains the elements common to both C and  D is called the intersection of C and D, written as

b) C ∪ D :

The set which contains all the elements of C and those of D is

called the union of C and D, written as

c) C ∩ not_D :

The set which contains the elements common to both C and  D is called the intersection of C and D, written as

d) not (C ∪D) :

e) not (C ∩D) :

9.By drawing Venn diagrams verify De Morgan’s laws

 A1 . not(A ∩B) =notA ∪ notB

The set which contains all the elements of A and those of B is called the union of A and B, but Not (A  and B) 

the set which not contains all the elements of A and those of B.

                                                                         

The set which contains all the elements of A and those of B is called the union of A and B, but Not (A  and B) 

the set which not contains all the elements of A and those of B.

So this answer : 

A2 .  not (A ∪B)  = notA ∩notB  
         
              not (A ∪B)   

                                                           

              notA ∩notB

                                                       

So not (A ∪B)  = notA ∩notB