7. Set operations

Set operations

Union

Set operation (Union)

The union of sets $A$ and $B$, $A\cup B$, is the set $$ A \cup B = \{ x | x \in A \text{ (inclusive)} \text{ or } x \in B \} $$

$A\cup B$ consists of elements in either $A$ or $B$ or both.

Venn diagram of $A\cup B$

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

Properties of union

• Identity law (Sets)

  The identity law is that $A\cup \varnothing =$ $A$.

  Link to original

• $A\subseteq B⟺A\cup B=B$

Link to original

Intersection

Set operation (Intersection)

The intersection of sets $A$ and $B$, $A\cap B$, is the set $$ A \cap B = \{ x | x \in A \text{ and } x \in B \} $$

$A\cap B$ consists of elements in both $A$ and $B$.

Venn diagram of $A\cap B$

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

Disjoint

Disjoint sets are sets with no common elements, $A \cap B = \emptyset$.

Link to original

Properties of intersection

• The domination law states that $A\cap \varnothing =\varnothing$.
• $A\subseteq B⟺A\cap B=A$

Link to original

Difference / Complement

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

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

Link to original

Absolute complement

Set operation (Absolute complement)

Sometimes we may consider all sets as being subsets of some given universal set, $U$.

Given a universal set $U$ and $A\subseteq U$, the complement of $A$, $\bar{A}$, is the set $$ \begin{aligned} \bar A &= U - A \\ &= \{ x \in U | x \notin A \} \end{aligned} $$

Venn diagram of $\bar{A}$

Link to original

Common Laws

• Commutative law (Sets)

  The commutative law says that $A\cup B=$ $B \cup A$ and $A \cap B = B \cap A$.

  Link to original

• Associative law (Sets)

  The associative law says that $(A\cup B)\cup C=A\cup (B\cup C)$ and that $(A\cap B)\cap C=A\cap (B\cap C)$.

  We can meaningfully write, $A\cup B\cup C$ or ${\bigcup }_{i=1}^n{A}_i$.

  Similarly, we can also write, $A\cap B\cap C$ or ${\bigcap }_{i=1}^n{A}_i$.

  Link to original

• Idempotent law (Sets)

  The idempotent law says that $A\cup A=$ $A$ and $A \cap A = A$.

  Link to original

• Distributive laws (Sets)

  There are two distributive laws: ?

  • $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
  • $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

  Link to original