3.3 UtU - sU:2 (Sets- Part 2):

 

pP: PɼV ĸAiģ UĤ:

MAz vgUwAi 60 zyU wAiƧg Pr Cx Q nA Cx Jgl nA jPƼ P. 45 A Pr nA jzg v 30 A Q nA jzg. Uzg Jgq nAU jPAq zyUɵ֔

UtU CzsAiģ EAv ĸU r AiPj.

 

3.3.1 UtU PtU (Properties of sets):

 

2+3 =3+2 , 2*3 =3*2.

AP v UuPgU jvAi.

(2+3)+4= 2+(3+4) ; (2*3)*4= 2*(3*4).

AP v UuPgU vAiĪV.

FU UtU PtU wAiĪ.

 

3.3.1 Gz 1 : UtU: A = {p,q,r,} ,B = {q,r,s,} v C={r,s,t} Dzg F PɼVU :

1.      BC =CB

2.      BC = CB

3.      A(BC) = (AB)C

4.      A(BC) = (AB) C

5.      A (BC) = (AB) (AC)

6.      A (BC) = (AB)(AC)

7.       

jg:

BC = {q,r,s}{r,s,t} = {q,r,s,t} ------(1)

CB = {r,s,t} {q,r,s} ={q,r,s,t} -------(2)

(1) v (2) jAz, BC =CB

1. UtU AAiU jvAiĪVz.(Union of sets is commutative):

BC = {q,r,s}{r,s,t} = {r,s} -----(3)

CB = {r,s,t} {q,r,s} = {r,s} -----(4)

(3) v (4) jAz, BC = CB

2. UtU bz jvAiĪVz.(Intersection of sets is commutative):

AB = {p,q,r,}{q,r,s} = {p,q,r,s}

A(BC) = {p,q,r} {q,r,s,t} ={p,q,r,s,t,} ---(5)

(AB)C= {p,q,r.s}{r,s,t} = {p,q,r,s,t} ---------(6)

(5) v (6) jAz, A(BC) = (AB)C

3. UtU AAiU vAiĪVz. (Union of sets is associative):

AB = {p,q,r}{q,r,s} = {q,r}

A (BC) ={p,q,r}{r,s} ={r} ------(7)

(AB) C = {q,r}{r,s,t} = {r} ------(8)

(7) v (8) jAz, A(BC) = (AB) C

4. UtU bz vAiĪVz.(Intersection of sets is associative):

A (BC) = {p,q,r}{r,s} = {p,q,r,s} -----------------(9)

AC = {p,q,r}{r,s,t} = {p,q,r,s,t}

(AB) (AC) = {p,q,r,s}{p,q,r,s,t} ={p,q,r,s} ----(10)

(9) v (10) jAz, A (BC) = (AB) (AC)

5.UtU AAiU bzz ï sdPvAiģ Az.(Union of sets is distributive over intersection of sets):

A (BC) = {p,q,r,}{q,r,s,t} ={q,r} ----(11)

(AB) = {p,q,r}{q,r,s} = {q,r}

(AC) = {p,q,r}{r,s,t} = {r}

(AB)(AC)= {q,r}{r} = {q,r} --------(12)

(11) v (12) jAz, A (BC) = (AB)(AC)

6.UtU bz AAiUz ï sdPvAiģ Az.(Intersection of sets is distributive over union of sets):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r iUߣ AiĪU(De Morgans laws):-

 

:

 

1. (AB)1= A1B1 (Jgq UtU AAiU Utz gPUt D Jgq UtU gP UtU bzP )

(The complement of union of sets is the intersection of their complements)

2. (AB)1= A1B1(Jgq UtU bz Utz gP Ut D Jgq UtU gP UtU AAiUP )

(The complement of the intersection of sets is the union of their complements)

 

3.3.1 Gz2 : U = {0,1,2,3,4,5,6,7,8,9} -- (10QAv PɼV ASU)A = {x:10QAv PɼV tU AS} B = {x:10QAv PɼV 3 g CvU}

FU A v B Utz UuAU gAiĪ.

A = {1,4,9} (Gz CAU tUU) B = {3,6,9} (3 = 3*1, 6=3*2,9=3*3)

A1 = U-A (AAiİ Ez U z Gz UuAU Ut A1)

= {0,1,2,3,4,5,6,7,8,9} - {1,4,9} ={0,2,3,5,6,7,8} =========(1)

B1= U-B (B Aiİ Ez U z Gz UuAU Ut B1)

={0,1,2,3,4,5,6,7,8,9} - {3,6,9} ={0,1,2,4,5,7,8} =========(2)

(1) v (2) jAz,

A1B1= {0,2,3,5,6.7,8}{0,1,2,4,5,7,8} ={0,2,5,7,8} ==================(3)

(AB) = {1,4,9}{3,6,9} = {1,3,4,6,9}

(AB)1 = U -(AB) = {0,1,2,3,4,5,6,7,8,9}- {1,3,4,6,9} = {0,2,5,7,8} ==(4)

(3) v (4) jAz,

1. (AB)1 = A1B1

(1) v (2) jAz

A1B1= {0,2,3,5,6,7,8}{0,1,2,4,5,7,8} ={0,1,2,3,4,5,6,7,8}==============(5)

AB = {1,4,9}{3,6,9}= {9}

(AB)1= U (AB) = {0,1,2,3,4,5,6,7,8,9}- {9} ={0,1,2,3,4,5,6,7,8} =======(6)

(5) v (6)jAz,

2. (AB)1 = A1B1

 

 

 

 

 

 

 

3.3.2 Jgq UtU UuAU ASU qī AAzs

(Relationship between numbers of elements of 2 sets)

 

A Utz UuAU ASAiģ n(A) JAz gAivê.(cardinal number )

 

3.3.2 Gz1 : A= {p,q,r,s,t} v B= {r,s,u,v,w} DVg.

n(A) =n(B)=5

AB ={p,q,r,s,t}{r,s,u,v,w}= {p,q,r,s,t,u,v,w}

AB ={p,q,r,s,t}{r,s,u,v,w} =(r,s} n(AB) =8, n(AB) =2

n(A) +n(B) = 5+5 =8+2 = n(AB) +n(AB)

F Pgt E ֪V U gAiħz:

1. n(AB)= n(A) +n(B)-n(AB)

2. n(AB)= n(A) +n(B)-n(AB)

3. A v B U AztPɬĮz UtUzg, n(AB)= n(A) +n(B)

( n(AB)=0 KPAzg AB = { }=(A v B U AztPɬĮz UtU).

 

3.3.2 ĸ1: M ƪigĪ Pɮ gU. 110 gU AU ƪUAz Prz, 50 gU İU ƪUz Prz. v 30 gU Jgq UAi ƪU Az. Uzg C EgĪ gU AS J?

 

jg:

A Ai AU UAz Prz gU UtVg. n(A) =110.

B Ai İU UAz Prz gU UtVg. n(B)= 50.

AB Ai Jgq UAi UAz Prz gU Ut. n(AB)=30.

AB Ai K gU Ut.

n(AB)= n(A) +n(B)-n(AB) = 110+50-30 =130

 

ƪigĪ 130 gU.

 

3.3.2 ĸ 2: MAz vgUwAi 60 zyU wAiƧg Pr Cx Q nA Cx Jgl nA jPƼ P. 45 A Pr nA jzg v 30 A Q nA jzg. Uzg Jgq nAU jPAq zyUɵ?( 3.1 pP Aiİ ĸ)

 

jg:

A Ai Pr nA zyU UtVg. n(A) =45

B Ai Q nA zyU UtVg. n(B) = 30

AB Ai Jgq nAU zyU UtVg.

n(AB)- PAqĻrAiĨPzz.

AB Ai vgUwAiİ zyU Ut.

n(AB)=60 - zv

n(AB)= n(A) +n(B)-n(AB) n(AB)= n(A) +n(B)- n(AB) = 45+30-60 =15

15 zyU Jgq nAU DAi Ezg.

 

3.3.2 ĸ 3 :MAz g PlAU AzV, 750 PlAU vZɯ 400 PlAU Qq Zɯ v 300 PlAU Jgq ZɯU QĪz PAqħAv.

Uzg.

1. J PlAU vZɯ iv Qvg?

2. J PlAU Qq Zɯ iv Qvg?

3. J PlAU n QĪŢ?

 

jg:

Azz PlAU Ut: U DVg. n (U) =1000

vZɯ qĪg PlAU Ut: A DVg. n (A) =750

Qq Zɯ qĪg PlAU Ut : B DVg. n(B)=400

Jgq Zɯ qĪg PlAU Ut: AB n(AB)=300

UĤ:

1. A-AB vZɯ iv qĪg Ut. CgĪ UuAU AS = n [A-AB].

2. B- AB Qq Zɯ iv qĪg Ut. CgĪ UuAU AS = n [B-AB].

3. AB Ai n qĪg Ut.

n(AB)= n(A)+n(B)-n(AB) = 750+400-300 = 850

4. (AB)1 mɰߣ qz EgĪg Ut. CgĪ UuAU AS = n(AB)1 FU,

1. n [A-AB] = n(A) n(AB) = 750 -300 = 450

2. n [B-AB] = n(B) n(AB) = 400 -300 = 100

3. n(AB)1= n[U (AB)] = n(U) n((AB)) = 1000-850 = 150

 

 

 

 

 

 

 

3.3 Pv SAU

 

 

P.A.

ɣqPz CAU

1

(AB)1 = A1B1

2

(AB)1 = A1B1

3

n(AB)= n(A) +n(B)-n(AB)

4

n(AB)= n(A) +n(B)-n(AB)