mth202 assingment no1

 

 

 Semster Spring 2023

MTH 202

Assignment No 1

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Question 1:

For any two sets A and B verify whether B- (B-A)= A∩B. Justify your answer with proper reasoning.

Solution:

LHS-B-(B-A)

B-(B∩A^C)                                Alter. Rep. for set difference

B-(B∩A^C)^C                              Alter. Rep. for set difference

B∩ (B^CU (A^C)^C)                                   DeMorgan's Law

B∩ (B^CUA)                              I Double Complementation Law

(B∩B^C) U (B∩A)                                 Distributive Law

φU(B∩A)                                 Complementary Law

B∩A                                        Identity Law

RHS A∩ B                               Proved

To verify whether B-(B-A)=A∩ B, let's break down each step of the solution and provide reasoning for the justification

By following the steps above, we can see that B-(B-A) simplifies to An B. Therefore, the expression B-(B-A) is equal to An B.

The reasoning behind each step is based on the fundamental laws and properties of set operations:

·         Set distinction (A-B) represents the elements which can be in Abutnotin B.

·         Rs.Morgan's Law states that the complement of the union/intersection of  sets is identical to the         intersection/union of their enhances, respectively.

·         Double Complementation Law states that the complement of a supplement of a fixed is the set itself.

·         Distributive Law permits us to distribute the intersection over the set distinction.

·         Complementary Law states that the intersection of a hard and fast and its complement is the empty    set

(l.E., the complement of a fixed includes all factors no longer within the set).

·         Identity Law states that the intersection of a set with the ordinary set (in this situation, B) is same to the unique set.By applying these laws and properties systematically, we can simplify the expression and verify that B-

(B-A) is equal to A∩B.

Question 2:

Find the matrix representing the relation where the matrices representing R and S are:

M_R =                    and   M_S =
Ans

To find the matrix representing the composition of relations R and S, denoted as SR, we need to perform matrix multiplication of the matrices representing R and S.

Given:

M_R =                    and   M_S =

Performing the matrix multiplication

 (1)(0)+(1)(0)+(0)(1) (1x0)+(1x1)+(0)(1) (1)(1)+(1)(0)+(0)1)

(1)(0)+(0)(0)+(1)α1) (1X0)+(0X1)+(1)(1) (1)(1)+(0)(0)+(1)(1)

(0) (0)+(0)(0)+(1)(1) (0X0)+(0X1)+(1X1) (0X1)+(0X0)+(1)(1)

Therefore, the matrix representing the composition of relations R and S, denoted as SR, is:

Simplifying the calculation:

=

Therefore the correct matrix SOR:

=