Decide Which Sets Are Subsets of Which
Decide among the following sets, which are subsets of which:
\[ A=\{x:x \text{ satisfies } x^2-8x+12=0\} \]
\[ B=\{2,4,6\} \]
\[ C=\{2,4,6,8,\ldots\} \]
\[ D=\{6\} \]
Solution
First solve \[ x^2-8x+12=0 \]
\[ x^2-6x-2x+12=0 \]
\[ x(x-6)-2(x-6)=0 \]
\[ (x-6)(x-2)=0 \]
Therefore, \[ x=2 \quad \text{or} \quad x=6 \]
Hence, \[ A=\{2,6\} \]
Now compare the sets:
Every element of \[ A=\{2,6\} \] is in \[ B=\{2,4,6\} \]
Therefore, \[ A \subseteq B \]
Also every element of \[ B \] is in \[ C \]
Therefore, \[ B \subseteq C \]
Since \[ D=\{6\} \] and \[ 6 \in A,B,C \]
Therefore, \[ D \subseteq A \]
\[ D \subseteq B \]
\[ D \subseteq C \]
Hence the subset relations are:
\[ D \subseteq A \subseteq B \subseteq C \]