Subsets

Let A = {$2$, $4$} and B = {$1$, $2$, $3$, $4$, $5$}. Is A ⊆ B?

• 1Check each element of A against B.
• 2Is $2$ in B? Yes.
• 3Is $4$ in B? Yes.
• 4Every element of A is in B, so A ⊆ B.

Answer: Yes, A ⊆ B

Let C = {$3$, $6$} and D = {$1$, $2$, $3$, $4$, $5$}. Is C ⊆ D?

• 1Check each element of C against D.
• 2Is $3$ in D? Yes.
• 3Is $6$ in D? No — $6$ is missing from D.
• 4Since one element fails, C is not a subset of D.

Answer: No, C ⊄ D

Let $G=\{1,3,5,7\}$ and $H=\{\{1\},\{3\},\{5\},\{7\}\}$.

• Is $\{1\}\subseteq G$? Yes — $1$ is an element of G, so the set containing only $1$ is a subset of G.
• Is $\{1\}\subseteq H$? Yes — $\{1\}$ is one of the elements listed inside H.
• Is $1\in H$? No — H's elements are sets like $\{1\}$, not the bare number $1$.
• Is $\varnothing \subseteq G$? Yes — the empty set is a subset of every set.
• Is $\{\}=\{\varnothing \}$? No — $\{\}$ has no elements; $\{\varnothing \}$ has one element (the empty set).