Joint sets

If two sets have elements in common, their circles overlap in a Venn diagram. These are called joint sets. For example:

$$A = \{a, b, c\}, \quad B = \{a, b, d\}$$

Sets A and B are joint because they share the elements $a$ and $b$.

Disjoint sets

If two sets have no elements in common, they are called disjoint sets. For example:

$$A = \{a, b\}, \quad B = \{1, 2\}$$

Subset relationships

If all elements of set $A$ are contained within set $B$, then $A \subseteq B$. This is shown in a Venn diagram where the circle for $A$ is entirely within the circle for $B$.

$$A = \{7, 8, 11\}, \quad B = \{6, 7, 8, 9, 10, 11, 12\}$$

a. Intersection of sets

The intersection $A \cap B$ is represented by the overlapping region of the two sets in the diagram.

b. Union of sets

The union $A \cup B$ is represented by the total shaded region covering all elements in either set.

c. Relative difference

The relative difference $A - B$ consists of the elements in $A$ but not in $B$. In a Venn diagram, this is the part of circle $A$ that does not overlap with $B$.

d. Symmetric difference

The symmetric difference $A \Delta B$ is defined as:

$$A \Delta B = (A - B) \cup (B - A)$$

This is shown in a Venn diagram as the shaded regions of $A \cup B$ excluding the intersection $A \cap B$.

e. Complement of a set

The complement of set $A$, denoted $A'$, contains all elements in the universal set $U$ that are not in $A$. In a Venn diagram, this is represented by shading the area of the rectangle outside circle $A$.

Use diagrams to visualize the following:

• $(P - M) - H$
• $(A \cup B) \cap (A \cup (A \cap B)')$
• $(A \cup B) \cap (A \cup B) \cap (A \cup B)$

Example universal set:

$$U = \{a, b, c, d, e\}$$

If $A = \{a, b, c\}$, then clearly $A \subseteq U$, and $A' = \{d, e\}$

Identity group 1

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

Identity group 2

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