Practice questions for 1rst year students of Eden Mohila College, Dhaka.

Subject:Mathematics for Economist;

Chapter: Concept of sets

Define(সঙ্গা লিখ) ঃ Null set(ফাঁকা সেট), Proper Sub set(প্রকৃত উপসেট),Power Set( শক্তি সেট),সংযোগ সেট,(Union of sets)Intersection of sets( ছেদক সেট),Equivalent set(সমতুল্য সেট),Universal set( সার্বিক সেট), Unique set(একক  সেট),  Disjoint set (পৃথকীকৃত সেট) , Partition of set(বিভক্ত সেট) ,Equal set(সমান সেট),Finite set(অসীম সেট,Set of Compliment(পরিপূরক সেট),Venn Euler’s diagram(ভেন ডায়াগ্রাম)

Define and prove the  laws of set  bellow:  

1.The Distributive Law

Distributive Law states that the sum and product remain the same value even when the order of the elements is altered.

First Law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Second Law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Problem: Let  A={1,2,3,4,5,6,7} , B={3,4,7} , C={2,3,5} ,Prove the Distributive Laws.

2.  The Associative 

  First law: The first law states that the intersection of a set to the intersection of two other sets is the same. (A ∩ B) ∩ C = A ∩ (B ∩ C)

Second Law: Second law states that the union of a set to the union of two other sets is the same. (A ∪ B) ∪ C = A ∪ (B ∪ C)

Problem: Let  A={1,2,3,4,5,6,7} , B={3,4,7} , C={2,3,5} ,Prove the  Associative  Laws

3. Commutative Law

Theorem 1 (Commutative Lεεaw for the Union of Two Sets): If A and B are  two finite sets then

          AB=BA.

Theorem 2 (Commutative Law for the Intersection of Two Sets): If A and B are sets then AB=BA.

4. De Morgan’s Law:

For any two finite sets A and B;

 (i) (A U B)’ = A’ ∩ B’ (De Morgan’s Law of Union).

 (ii) (A ∩ B)’ = A’ U B’ (De Morgan’s Law of Intersection).

5. Justify the . De Morgan’s Law   

1.Let U = {j, k, l, m, n}, X = {j, k, m} and Y = {k, m, n}. Prove : (X ∩ Y)’ = X’ U Y’

Solution:

Here, U = {j, k, l, m, n} X = {j, k, m} Y = {k, m, n}

Step 1:

(X ∩ Y) = {j, k, m} ∩ {k, m, n} = {k, m} Therefore, (X ∩ Y)’ = {j, l, n} ……………….. (i) X = {j, k, m} Therefore, X’ = {l, n} Y = {k, m, n} Therefore, Y’ = {j, l}

Step 2:

X’ ∪ Y’ = {l, n} ∪ {j, l} Therefore, X’ ∪ Y’ = {j, l, n} ……………….. (ii) Combining (i) and (ii) we get, (X ∩ Y)’ = X’ U Y’ Hence, the De Morgan’s Law is Proved

2.Let U = {1, 2, 3, 4, 5, 6, 7, 8}; P = {4, 5, 6} and Q = {5, 6, 8}. Prove De Morgan’s Law (P ∪ Q)’ = P’ ∩ Q’.

Solution:

Here, U = {1, 2, 3, 4, 5, 6, 7, 8} P = {4, 5, 6} Q = {5, 6, 8}

Step 1:

P ∪ Q = {4, 5, 6} ∪ {5, 6, 8} = {4, 5, 6, 8} Therefore, (P ∪ Q)’ = {1, 2, 3, 7} ……………….. (i) P = {4, 5, 6} Therefore, P’ = {1, 2, 3, 7, 8} Q = {5, 6, 8} Therefore, Q’ = {1, 2, 3, 4, 7}

Step 2:

P’ ∩ Q’ = {1, 2, 3, 7, 8} ∩ {1, 2, 3, 4, 7} Therefore, P’ ∩ Q’ = {1, 2, 3, 7} ……………….. (ii) Combining (i) and (ii) we get; (P Q)’ = P’ ∩ Q’ Hence, De Morgan’s Law is proved.

 

6.Idempotent Laws:

For any finite set A;

(i) A U A = A

(ii) A ∩ A = A

7.Identity law:

For any finite set A, Null set Ø and universal set U:

(i) A U Ø  = A

(ii) A ∩ U = A

Some problems:

PROBLEM-1 : Let  A={1,2,3,4} , B={2,4,6,8} , C={3,4,5,6} ,             

Find (1) A UB , (2) A UC , (3) BUC (4) BUB (5) (AUB)UC (6) AU(BUC)

PROBLEM-2: Let X=={Sakib,Musfik,Mashrafi} , Y={Sakib,Mustafij,Sommo} , Z={ Mustafij,Sommo ,Tamim} .

Find(1)XUY ,(2) YUZ (3) XUZ

PROBLEM-3 : Let  A={1,2,3,4} , B={2,4,6,8} , C={3,4,5,6} ,Find

(i) A ∩ B  , (2) A ∩C , (3) B∩C (4) B∩B (5) (A∩B) ∩C (6) A∩ (B∩C)

PROBLEM-4 : Let  A={1,2,3,4,5,6,7} , B={3,4,7} , C={2,3,5} ,

Find (1) A -B , (2) A UC , (3) C-A (4) B-C (4)B-A  (5)B-B (6) A-(BUC)

(7) (A-B)U(A-C)

PROBLEM-5: Let U=  {1,2,3,4,5,6,7,8,9} , A={1,2,3,4}  B={2,4,6,8} , C={3,4,5,6} ,

Find (1) A, (2) B , (3) (A ∩C) (4) (A UB) (5) (A) (6) (B-C)

PROBLEM-6:Shade the problem by Venn diadram: (1) A, (2) (AUB) , (3) (B-A)

 (4) A∩B‘(5)  A ∩( BUC) (6) ) (A ∩ B)U( A ∩ C) (7) AU(B∩C) (8) (A U B)∩(AUC)

PROBLEM-7: U=={a,b,c,d,e} , A={a,b,d} , B={b,d,e} ,Find

(i) A U B  , (2) B∩A , (3) B (4) B-A (5) (A∩B)  (6) AUB‘ (7)A’∩B‘ (8)B’-A’ (9) ) (AUB )‘(10) (A∩B)  .

PROBLEM-8: U=={a,b,c,d,e,f,g} , A={a,b,d,e,f} , B={a,c,e,g} ,C= B={b,e,f,g} .  Find

(i) A U C  , (2) B∩A , (3) C-B (4)  B  (5) A-B  (6) B‘UC (7) (A-C)’ (8)C‘A (9) ) (A-B )‘(10) (A∩A)

PROBLEM-9:If R= {x,y,z}.How many subsets of set R contain? And what are they?

PROBLEM-10:If  M={r,s,t}.What will be the power set of A? i.e P(A)=?

PROBLEM-11: Which of these sets are equal: {r,t,s},{s,t,r,s},{t,s,t,r},{s,r,s,t}?

PROBLEM-12: Which of the following sets are different? ∅,{0} {}

PROBLEM-13:In a class,85 students play Football,72 students play Cricket,& 43 students play Tennis.44 students play both Football & Cricket,15 play both Cricket and Tennis.25 students play both Football and Tennis.4 students play all 3 sports.

Find:

1)Total no. of students in class.

2)No. of students that play Cricket & Tennis but no Football.

3)No.  of students that play only Football.

Related Posts

মন্তব্য করুন

আপনার ই-মেইল এ্যাড্রেস প্রকাশিত হবে না। * চিহ্নিত বিষয়গুলো আবশ্যক।

bn_BDBengali
en_USEnglish bn_BDBengali