Sets

Sets are a fundamental concept in mathematics used to group elements that share common characteristics. They are represented using curly brackets \(\left\lbrace\right\rbrace\) and can contain numbers, letters, geometric figures, or any well-defined objects.

Set Notation

1. Roster Form (List Notation): Lists all elements inside curly brackets. Example: \( A = \left\lbrace 1,2,3,4,5 \right\rbrace\).
2. Descriptive Form: Describes the property that the elements satisfy. Example: \( B = \left\lbrace x \vert x \mathrm{\ is\ an\ even\ number\ less\ than\ } 10 \right\rbrace \).
3. Set-Builder Notation: Uses a mathematical expression to define the set. Example: \( C=\left\lbrace x \in \mathbb{N} \vert x < 5 \right\rbrace \) (Set of natural numbers less than 5).

Properties of Sets

• Membership: Denoted by \(\in\), meaning that an element belongs to a set. Example: \( 3 \in A \), \( 7 \notin A \)
• Subset: Denoted by \(\subseteq\), meaning that all elements of one set are also in another. Example: \( B \subseteq A \) if all elements of \(B\) are in \(A\).
• Empty Set: Represented as \(\empty\) or \( \left\lbrace\right\rbrace \), it has no elements.
• Universal Set: Represented by \(U\), it contains all elements under consideration in a given context.
• Equality of Sets: Two sets are equal if they have exactly the same elements.

Operations with Sets

1. Union \( \left( \cup \right) \)
• The set of elements that are in at least one of the sets.
• Example: \(A \cup B = \left\lbrace 1,2,3,4,5,6,8 \right\rbrace \)
2. Intersection \( \left( \cap \right) \)
• The set of elements that are in both sets.
• Example: \(A \cap B = \left\lbrace 2,4 \right\rbrace \)
3. Difference \( \left( - \right. \) or \( \left. \backslash \right) \)
• The set of elements that are in one set but not in the other.
• Example: \( A - B = \left\lbrace 1,3,5 \right\rbrace \)
4. Complement \( \left( A^{\complement} \right. \) or \( A^{\prime} \Big) \)
• The set of elements in the universal set \(U\) that are not in \(A\).
5. Symmetric Difference \( \left( \triangle \right) \)
• The elements that are in one of the sets but not in both.
• Example: \( A \triangle B = (A - B) \cup (B - A) \)

Venn Diagrams

Venn diagrams are graphical representations used to visualize the relationship between sets. They are usually drawn with overlapping circles inside a rectangle that represents the universal set. Each circle represents a set, and the overlapping regions show the elements that the sets have in common. These diagrams are useful for explaining operations such as union, intersection, complement, and difference of sets because they help organize information in a clear and visual way.

Interactive Venn diagram in GeoGebra

Click on the boxes numbered \(1-8\) to color their respective surfaces. When you think your answer is complete, click on the "Check your Venn diagram!" box. If your answer is correct, the word "Correct" will appear. If you want a new exercise, click on the yellow "New exercise" button.

Set explorer

Use this interactive explorer to enter the universal set and two working sets. The calculator updates operations, relationships, Venn regions, Cartesian products, and quick exercises directly in the browser.

See also

Bayes Theorem

Probability distributions formula sheet