Set operation (Difference)

The difference of sets $A$ and $B$, $A-B$, is the set $$ A - B = \{ x | x \in A \text{ and } x \notin B \} $$

$A-B$ consists of those elements that are in $A$ but not in $B$. This can also be called the complement of $B$ with respect to $A$.

Venn diagram of $A-B$

For example, let $A=\{4,7,8\}$ and $B=\{10,4,9\}$. Hence $A-B=\{7,8\}$ and $B-A=\{10,9\}$.

Properties of complementation

• Complementation laws (Sets)

  There are complementation laws that say that:

  • $A\cup \bar{A}=U$
  • $A\cap \bar{A}=\varnothing$
  • $\overline{\stackrel{ˉ}{A}}=A$
  • $\bar{U}=\varnothing$
  • $\bar{\varnothing }=U$

  Link to original

• De Morgan's law (Sets)

  The De Morgan’s law can be represented in sets as so: ?

  • $\overline{A\cup B}=\bar{A}\cap \bar{B}$
  • $\overline{A\cap B}=\bar{A}\cup \bar{B}$

  Link to original

• $A-B=A\cap \bar{B}$
• $A\subseteq B⟺\bar{B}\subseteq \bar{A}$