If all of the members of set LaTeX: A are all contained in the set LaTeX: B, then LaTeX: A is a subset of LaTeX: B, written as LaTeX: A\subset B. It is important to note that

  • the empty set, LaTeX: \varnothing or LaTeX: \left\{\right\}, is a subset of every set,
  • the set itself is a subset of the set,
  • these are LaTeX: 2^n subsets in every set, where n represents the number of elements in the set.

Given that LaTeX: F=\left\{f,\:a,\:s,\:t\right\}, LaTeX: G=\left\{g,\:a,\:s\right\} and LaTeX: S=\left\{s,\:t,\:a,\:f,\:f,\:r,\:o,\:o,\:m\right\} we can state that
LaTeX: F\subset S since all of the elements in LaTeX: F and contained in LaTeX: S
LaTeX: G\not\subset S since the element LaTeX: g in LaTeX: G is not a member of LaTeX: S
There are LaTeX: 2^3=2\times2\times2=8 subsets in LaTeX: G. These subsets are
LaTeX: \varnothing, LaTeX: \left\{g\right\}, LaTeX: \left\{a\right\}, LaTeX: \left\{s\right\}, LaTeX: \left\{a,\:g\right\}LaTeX: \left\{a,\:s\right\}, LaTeX: \left\{g,\:s\right\}, LaTeX: \left\{g,\:a,\:s\right\}