Question

Decide, among the following sets, which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x 2 – 8 x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

Answer 1

The sets are given as,

A={ x:x∈R and x satisfy  x 2 −8x+12=0 }

B={ 2,4,6 }

C={ 2,4,6,8... }

D={ 6 }

Solve the quadratic equation of set A to find its elements.

x 2 −8x+12=0 x 2 −6x−2x+12=0 x( x−6 )−2( x−6 )=0

Solve the above equation to find the roots of x.

x=2,6

Thus,

A = { 2,6 }

Set A is the subset of set B, if all the elements of A are in B.

For set A,

All the elements of set A are in set B and C.

Therefore, A is a subset of B, that is,

A ⊂ B

And, Ais a subset of C, that is,

A ⊂ C

For set B,

All the elements of set B are in set C.

Therefore, B is a subset of C, that is,

B ⊂ C

For set C,

All the elements of set C are not in any set.

Therefore, C is not a subset of any set.

For set D,

All the elements of set D are in set A, B and C.

Therefore,

D is a subset of A, that is,

D ⊂ A

D is a subset of B, that is,

D ⊂ B

And, D is a subset of C, that is,

D ⊂ C