3. If U = {a, e, i, o, u}, A = {a, e, i}, B = {e, o, u},
C = {a, i, u), then :
(i) What is A UU?
(ii) What is A U U?
(iii) What is AU ∅?
(iv) What is A Interaction ∅?
(v) Verify that A N (B – C) = (A intersection B) – (A InteractionC).
(vi) Verify that A – (B U C) = (A – B) interaction (A – C).
(vii) Verify that A – (B interaction C) = (A – B) U (A – C).
Answer:
aujla ni aujla
Step-by-step explanation:
lobe you meri siso jaan
(i) A UU = {a,e,i,o,u}
(ii) A U U = {a,e,i,o,u}
(iii) AU ∅= {a, e, i}
(iv) A Interaction ∅= ∅
(v) s A intersection (B-C) = (A intersection B)-(A intersection C)
Two sets are equal if both are subsets of each other.
Let x∈A∩(B−C).
⇒x∈A and x∈(B−C)⇒x∈B and x∉C.
⇒x∈A∩B and x∉A∩C⇒x∈A∩B−A∩C.
⇒A∩(B−C)⊂A∩B−A∩C.
Let x∈A∩B−A∩C.
⇒x∈A∩B and x∉A∩C.
⇒x∈A,x∈B and x∉C.
⇒x∈A and x∈B−C.
⇒x∈A∩(B−C).
⇒A∩B−A∩C⊂A∩(B−C).
⇒A∩(B−C)=A∩B−A∩C.
(vi) Let A, B, and C be three sets. Prove that A-(BUC) = (A-B) ∩ (A-C)
L.H.S = A – (B U C)
A ∩ (B U C)c
A ∩ (B c ∩ Cc)
(A ∩ Bc) ∩ (A∩ Cc)
(AUB) ∩ (AUC)
R.H.S = (A-B) ∩ (A-C)
(A∩Bc) ∩ (A∩Cc)
(AUB) ∩ (AUC)
L.H.S = R.H.S
(vii) Let x in A ∩ (B U C)
Then x is in A and x is in (B U C).
If x is in B, then x is in A ∩ B.
If x is not in B, then x is in C, so x is in A ∩ C.
Thus x is in (A ∩ B) U (A ∩ C), and A ∩ (B U C) ⊆ (A ∩ B) U (A ∩ C).
Now assume x in (A ∩ B) U (A ∩ C) and similarly show that x is in A ∩ (B U C).
Then (A ∩ B) U (A ∩ C) ⊆ A ∩ (B U C).
So (A ∩ B) U (A ∩ C) = A ∩ (B U C)
hope it will help u ✨☺️