Decide among the following sets , which are subsets of which :
A = {x : x satisfies x2−8x+12=0},
B = {2, 4, 6}, C = {2, 4, 6, 8, .....}, D = {6}.
We have,
A = {x : x satisfies x2−8x+12=0}
= {x : x2−6x−2x+12=0}
= {x : x (x-6)- 2(x-6)=0}
= {x : (x-6)(x-2)=0}
={x :x = 6, 2}
= {6, 2}
B = {2, 4, 6}
C = {2, 4, 6, 8, ....}
D = {6}
We know that if E and F are two sets, then E is a subset of F, i.e., E ⊆ F if x ϵ E ⇒ x ~\epsilon~F. E is called a proper subset of F if E is strictly contained in F and is denoted by E ⊂ F.
Clearly,
D ⊂ A { ∵ 6 ϵ A}
A ⊂ B {∵ 2, 6 ϵ A and they also belong to B}
Similarly, B ⊂ C
Hence, D ⊂ A ⊂ B ⊂ C.