Question

Let B;= [0,+ 1], for each integer i=1,2,3,4. Answer the following parts, showing all work. a) Find By U B2 U B3 U B4 b) Find Bin B2 B3 B4 c) Are the sets Bj, i=1,2,3,4, mutually disjoint? EXPLAIN. TTT Arial 3 (12pt) AT := - = - See's Path:p Words:0

Answer 1

Ans: Given that, Bi=[0,i+1] , for each integer i=1,2,3,4.

Then for i=1,  B1= [0, 1+1] =[0,2]

    i=2,  B2= [0, 2+1] =[0,3]

i=3,  B3= [0, 3+1] =[0,4]

i=4,  B4= [0, 4+1] =[0,5]

(a) Now we need to find B1$\cup$B2$\cup$B3$\cup$B4

We know, x∈(A∪B) if and only if (x∈A) or (x∈B)

So, B1$\cup$B2$\cup$B3$\cup$B4 = (((B1$\cup$B2)$\cup$B3)$\cup$B4) = (([0,2]$\cup$[0,3])$\cup$[0,4])$\cup$[0,5])

=(([0,2,3]$\cup$[0,4])$\cup$[0,5])

=([0,2,3,4]$\cup$[0,5])

=[0,2,3,4,5]

(b) Now we need to find B1$\cap$B2$\cap$B3$\cap$B4

We know, x∈(A$\cap$B) if and only if (x∈A) and (x∈B)

So, B1$\cap$B2$\cap$B3$\cap$B4 = [0,2]$\cap$[0,3]$\cap$[0,4]$\cap$[0,5]

= [0] Since there is only one common point among all the four sets.

(c) We know,Disjoint set are those sets whose intersection with each other results in a null set.In other words, if F is a family of sets, the sets in F are mutually disjoint if and only if Ai ∩ Aj = ∅  for all Ai, Aj ∈ F and i $\neq$ j .

We can see from the above  B1$\cap$B2$\cap$B3$\cap$B4 = [0] $\neq$ ∅ .

Hence the sets Bi i=1,2,3,4 are not mutually disjoint.   

Ans.