Basic operations

Subsection 2.2.1 Basic operations

Definition 2.2.1.1.

Let $A$ and $B$ be sets. The union of sets $A$ and $B\text{,}$ denoted $A \cup B$ is the set that contains those elements that are in $A$ or in $B$ or in both. That is,

\begin{equation*} A \cup B = \set{x \given x \in A\lor x \in B}. \end{equation*}

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Example 2.2.1.2. Union.

Let $A = \set{1, 2, 3, 5, 10}\text{,}$ and let $B = \set{2, 3, 4, 5}\text{.}$ To compute the union $A \cup B\text{,}$ we combine all of the elements of $A$ together with all of the elements of $B$ into one set. The union of $A$ and $B$ is

\begin{equation*} A \cup B = \set{1, 2, 3, 4, 5, 10}. \end{equation*}

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Inside a box that represents the universal set U there is an oval that represents the set A and an oval that represents the set B. The two ovals overlap. The interior of the two ovals, including the area where they overlap, is shaded. — Figure 2.2.1.3. The shaded area in the Venn diagram corresponds to the union $A \cup B$ of two sets $A$ and $B\text{.}$
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Definition 2.2.1.4.

Let $A$ and $B$ be sets. The intersection of $A$ and $B\text{,}$ denoted $A \cap B$ is the set that contains those elements that are in $A$ and also in $B\text{.}$ That is,

\begin{equation*} A \cap B = \set{x \given x \in A\land x \in B}. \end{equation*}

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Example 2.2.1.5. Intersection.

Let $A = \set{1, 2, 3, 5, 10}\text{,}$ and let $B = \set{2, 3, 4, 5}\text{.}$ To compute the intersection $A \cap B\text{,}$ we take the elements of $A$ that are also elements of $B\text{.}$ In other words, we only keep elements that $A$ and $B$ have in common. The intersection of $A$ and $B$ is

\begin{equation*} A \cap B = \set{2, 3, 5}. \end{equation*}

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Definition 2.2.1.7.

The difference of $A$ and $B\text{,}$ denoted $A - B\text{,}$ is the set containing elements that are in $A$ but not in $B\text{.}$ That is,

\begin{equation*} A - B = \set{x \given x \in A\land x\not \in B}. \end{equation*}

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Example 2.2.1.8. Two differences.

Let $A = \set{1, 2, 3, 5, 10}\text{,}$ and let $B = \set{2, 3, 4, 5}\text{.}$ The difference of $A$ and $B$ consists of members of $A$ that are not also members of $B\text{,}$ so we take all the elements of $A$ and throw out the ones that are in $B\text{.}$ That gives

\begin{equation*} A - B = \set{1, 10}. \end{equation*}

Analogously, the difference of $B$ and $A$ consists of members of $B$ that are not also members of $A\text{,}$ so

\begin{equation*} B - A = \set{4}. \end{equation*}

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Definition 2.2.1.11.

The symmetric difference of $A$ and $B\text{,}$ denoted by $A\lxor B\text{,}$ is the set containing those elements in either $A$ or $B\text{,}$ but not in both $A$ and $B\text{.}$ That is

\begin{equation*} A\lxor B = \set{x \given x \in A\lxor x \in B}. \end{equation*}

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Example 2.2.1.12. Symmetric difference.

Let $A = \set{1, 2, 3, 5, 10}\text{,}$ and let $B = \set{2, 3, 4, 5}\text{.}$ To compute the symmetric difference of $A$ and $B\text{,}$ we combine the elements of $A$ and $B\text{,}$ and them remove the elements that are members of both $A$ and $B\text{.}$ Thus

\begin{equation*} A\lxor B = \set{1, 4} \text{ and } B\lxor A = \set{1, 4}. \end{equation*}

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Definition 2.2.1.14.

Let $U$ be the universal set. The complement of $A\text{,}$ denoted $\overline{A}$ , is the difference of $U$ and $A\text{.}$ That is

\begin{equation*} \overline{A} = \set{x \given x \not \in A} = U - A. \end{equation*}

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Example 2.2.1.15. Complement.

Let $A = \set{1, 2, 3, 5, 10}\text{,}$ and let $B = \set{2, 3, 4, 5}\text{.}$ Suppose $U = \set{1, 2, \dots,10}\text{.}$ Then the complement of $A$ consists of elements in $U$ that are not in $A\text{.}$ We can compute it by taking all of the elements of $U$ and throwing out the elements of $A\text{.}$ That gives

\begin{equation*} \overline{A} = \set{4, 6, 7, 8, 9}. \end{equation*}

Analogously, the complement of $B$ is consists of members of $U$ that are not in $B\text{,}$ so

\begin{equation*} \overline{B} = \set{1, 6, 7, 8, 9, 10}. \end{equation*}

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Inside a box that represents the universal set U there is an oval that represents the set A. All of the interior of the box representing U is shaded, except for the area inside the oval representing B. — Figure 2.2.1.16. The shaded area in the Venn diagram corresponds to the complement $\overline{A}$ of the set $A\text{.}$
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Definition 2.2.1.17.

Two sets $A$ and $B$ are disjoint if $A \cap B = \emptyset\text{.}$

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Example 2.2.1.18. Disjoint.

Let $A = \set{1,3,5,7}\text{,}$ and let $B = \set{2, 4, 6}\text{.}$ Then $A$ and $B$ have no elements in common, so $A \cap B = \emptyset\text{.}$ Thus $A$ and $B$ are disjoint.

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Example 2.2.1.19. Intersection with a subset.

Prove if $A \subseteq B\text{,}$ then $A \cap B = A\text{.}$

Proof.

We need to show

$A \cup B \subseteq B\text{,}$ and
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$B \subseteq A \cup B\text{.}$
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The second statement $B \subseteq A \cup B$ is clear from the the definition. Namely, for any element $b$ in $B\text{,}$ we must have $b \in A \cup B$ by definition of union. It remains to show $A \cup B \subseteq B\text{.}$ Let $x$ be a generic element in $A \cup B\text{.}$ Then by definition, $x \in A$ or $x \in B\text{.}$ Since $A \subseteq B\text{,}$ in either case we have $x \in B\text{.}$ Thus $A \cup B \subseteq B\text{.}$

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Therefore $A \cup B = B$ as desired.

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Example 2.2.1.20. Venn diagram for $(A - B) \cup (B - A)\text{.}$.

Draw a Venn diagram representing $(A - B) \cup (B - A)\text{.}$

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Answer.

$Inside a box that represents the universal set U there is an oval that represents the set A and an oval that represents the set B. The two ovals overlap. The interior of oval representing the sets A abd B, excluding the area the two ovals overlap, is shaded.$

Comparing this Venn diagram with Figure 2.2.1.13 we notice that $(A - B) \cup (B - A)=A\oplus B\text{.}$

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Solution.

From the definition, $(A - B) \cup (B - A)$ consists of elements of $A$ that are not in $B$ together with elements of $B$ that are not in $A\text{.}$ We shade the part of $A$ that is not in $B$ and the part of $B$ that is not in $A\text{.}$

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Example 2.2.1.21. Venn diagram for $A \cap (B \cup C)\text{.}$.

Draw a Venn diagram representing $A \cap (B \cup C)\text{.}$

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Answer.

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Solution.

From the definition, we have that $A \cap (B \cup C)$ consists of elements that are in $A$ and in $B \cup C\text{.}$

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Example 2.2.1.22. Venn diagram for $\overline{A} \cap \overline{B} \cap \overline{C}\text{.}$.

Draw a Venn diagram representing $\overline{A} \cap \overline{B} \cap \overline{C}\text{.}$

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Answer.

$Inside a box that represents the universal set U there are three ovals that represent sets A, B, and C. The three sets overlap. This partitions U into eight regions. Three regions are shaded: the region that is in A, B, and C; the region that is in A and B, but not in C; the region that is in A and C, but not in B.$

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Solution.

From the definition, we have that $\overline{A} \cap \overline{B} \cap \overline{C}$ consists of elements that are not in $A$ and not in $B$ and not in $C\text{.}$

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