Homework Answers
Proof: (A U B)∩C =(A∩C) U (B∩C)
Mathematics we use the following symbols:
U = OR = V
∩ = AND = ∧
The law state that taking intersection of a set to union of two other set is the same as taking the intersection of orginal set and both two sets separately and then taking union of the result.
let x 
(AUB)∩C ,
if x (AUB)∩C
then x is either in A or in(B and C).
x (A orB )
and x C
{x A or x
B} and x
C
{x A and x
C} or { x
B and x
C}
x (A and C)
or x (B and
C)
x
(A∩C)U(B∩C)
therefore,
(A U B)∩C ⊂ (A∩C) U (B∩C) .............1
let x (A∩C)U(B∩C)
, if x (A∩C)U(B∩C)
then x is in (A and B) or x is in (B and C).
x (A and C)
or x (B and
C)
{x A
and x C} or {x
B and x
C}
{x A or x
B} and x
C
{ x (AUB)} ∩ x
C
x (AUB)∩C
therrfore (A∩C)U(B∩C) ⊂ (A U B)∩C ...........2
from equation 1 and 2
(A U B)∩C =(A∩C) U (B∩C)
Proof: (A U B)∩C =(A∩C) U (B∩C)
Mathematics we use the following symbols:
U = OR = V
∩ = AND = ∧
The law state that taking intersection of a set to union of two other set is the same as taking the intersection of orginal set and both two sets separately and then taking union of the result.
let x 
(AUB)∩C ,
if x (AUB)∩C
then x is either in A or in(B and C).
x (A orB )
and x C
{x A or x
B} and x
C
{x A and x
C} or { x
B and x
C}
x (A and C)
or x (B and
C)
x
(A∩C)U(B∩C)
therefore,
(A U B)∩C ⊂ (A∩C) U (B∩C) .............1
let x (A∩C)U(B∩C)
, if x (A∩C)U(B∩C)
then x is in (A and B) or x is in (B and C).
x (A and C)
or x (B and
C)
{x A
and x C} or {x
B and x
C}
{x A or x
B} and x
C
{ x (AUB)} ∩ x
C
x (AUB)∩C
therrfore (A∩C)U(B∩C) ⊂ (A U B)∩C ...........2
from equation 1 and 2
(A U B)∩C =(A∩C) U (B∩C)
Proof: (A U B)∩C =(A∩C) U (B∩C)
Mathematics we use the following symbols:
U = OR = V
∩ = AND = ∧
The law state that taking intersection of a set to union of two other set is the same as taking the intersection of orginal set and both two sets separately and then taking union of the result.
let x 
(AUB)∩C ,
if x (AUB)∩C
then x is either in A or in(B and C).
x (A orB )
and x C
{x A or x
B} and x
C
{x A and x
C} or { x
B and x
C}
x (A and C)
or x (B and
C)
x
(A∩C)U(B∩C)
therefore,
(A U B)∩C ⊂ (A∩C) U (B∩C) .............1
let x (A∩C)U(B∩C)
, if x (A∩C)U(B∩C)
then x is in (A and B) or x is in (B and C).
x (A and C)
or x (B and
C)
{x A
and x C} or {x
B and x
C}
{x A or x
B} and x
C
{ x (AUB)} ∩ x
C
x (AUB)∩C
therrfore (A∩C)U(B∩C) ⊂ (A U B)∩C ...........2
from equation 1 and 2
(A U B)∩C =(A∩C) U (B∩C)
4. Let A, B, and C be sets. Prove that AU(BNC) = (AUB) n (AUC).
4. Let A, B, and C be sets. Prove that AU(BNC) = (AUB) n (AUC).
write the proof problem 3 2. Let A, B and C be sets, then Au(Bnc)-(AUB)n (Auc)...
write the proof problem 3
2. Let A, B and C be sets, then Au(Bnc)-(AUB)n (Auc) 3. Let A and B be sets, then (An B)c-AcUBc.
3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB)...
3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB) P(A)P(B), (b) P(AUBUC) P(A)+P(B)+P(C)-P(AnB)-P(Anc)-P(Bnc)+P(AnBnc)
Please help me prove 2,4, and 5. Thank you Theorem 17. Let A, B and C...
Please help me prove 2,4, and 5. Thank you
Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB)...
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)
2.4. Prove AN(BUC) -An B)U(ANC) P(AUB) = PCA)+PCB)- P(ANB)
2.4. Prove AN(BUC) -An B)U(ANC) P(AUB) = PCA)+PCB)- P(ANB)
61. Prove or sets A, B, and C disprove: A A (BnC) (A A B) n...
61. Prove or sets A, B, and C disprove: A A (BnC) (A A B) n (A A C) for all
Exercises for Chapter 17 1. (a) Let A, B be sets. Prove that AUB=A if and...
Exercises for Chapter 17 1. (a) Let A, B be sets. Prove that AUB=A if and only if B CA.
A,C,G please 1. Let A, B, and C be subsets of some universal set u. Prove...
A,C,G please
1. Let A, B, and C be subsets of some universal set u. Prove the following statements from Theorem 4.2.6 (a) AUA=/1 and AnA=A. (b) AUO- A and An. (c) AnB C A and ACAUB (d) AU(BUC)= (A U B) U C and An(B n C)-(A n B) n C. (e) AUB=BUA and A n B = B n A. (f) AU(BnC) (AU B) n(AUC) (g) (A U B) = A n B (h) AUA=1( and An-=0. hore...
6. Let A, B, and C be subsets of some universal set U. Prove or disprove each of the following: *...
6. Let A, B, and C be subsets of some universal set U. Prove or disprove each of the following: * (a) (A n B)-C = (A-C) n (B-C) (b) (AUB)-(A nB)=(A-B) U (B-A)
6. Let A, B, and C be subsets of some universal set U. Prove or disprove each of the following: * (a) (A n B)-C = (A-C) n (B-C) (b) (AUB)-(A nB)=(A-B) U (B-A)
4. Let A, B, and C be sets. Prove that AU(BNC) = (AUB) n (AUC).
write the proof problem 3
2. Let A, B and C be sets, then Au(Bnc)-(AUB)n (Auc) 3. Let A and B be sets, then (An B)c-AcUBc.
3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB) P(A)P(B), (b) P(AUBUC) P(A)+P(B)+P(C)-P(AnB)-P(Anc)-P(Bnc)+P(AnBnc)
Please help me prove 2,4, and 5. Thank you
Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)
2.4. Prove AN(BUC) -An B)U(ANC) P(AUB) = PCA)+PCB)- P(ANB)
61. Prove or sets A, B, and C disprove: A A (BnC) (A A B) n (A A C) for all
Exercises for Chapter 17 1. (a) Let A, B be sets. Prove that AUB=A if and only if B CA.
A,C,G please
1. Let A, B, and C be subsets of some universal set u. Prove the following statements from Theorem 4.2.6 (a) AUA=/1 and AnA=A. (b) AUO- A and An. (c) AnB C A and ACAUB (d) AU(BUC)= (A U B) U C and An(B n C)-(A n B) n C. (e) AUB=BUA and A n B = B n A. (f) AU(BnC) (AU B) n(AUC) (g) (A U B) = A n B (h) AUA=1( and An-=0. hore...
6. Let A, B, and C be subsets of some universal set U. Prove or disprove each of the following: * (a) (A n B)-C = (A-C) n (B-C) (b) (AUB)-(A nB)=(A-B) U (B-A)
6. Let A, B, and C be subsets of some universal set U. Prove or disprove each of the following: * (a) (A n B)-C = (A-C) n (B-C) (b) (AUB)-(A nB)=(A-B) U (B-A)
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