Unions Intersections Complements Calculator

Set Operations:

\[ \begin{align*} A \cup B &= \{x \mid x \in A \text{ or } x \in B\} \\ A \cap B &= \{x \mid x \in A \text{ and } x \in B\} \\ A^c &= \{x \mid x \notin A\} \end{align*} \]

Union (A ∪ B):

Intersection (A ∩ B):

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1. What Are Set Operations?

Set operations are fundamental operations in mathematics that combine or relate sets in various ways. The three primary operations are union (∪), intersection (∩), and complement (^c).

2. How Does the Calculator Work?

The calculator performs the following operations:

\[ \begin{align*} A \cup B &= \{x \mid x \in A \text{ or } x \in B\} \\ A \cap B &= \{x \mid x \in A \text{ and } x \in B\} \\ A^c &= \{x \mid x \notin A\} \end{align*} \]

Where:

\( A \cup B \) — All elements that are in A, or in B, or in both
\( A \cap B \) — Only elements that are in both A and B
\( A^c \) — All elements not in A (relative to a universal set)

3. Importance of Set Operations

Details: Set operations are fundamental in mathematics, computer science, database theory, and many other fields. They form the basis for logical operations, database queries, and probability theory.

4. Using the Calculator

Tips: Enter elements of your sets separated by commas. For complement operations, you must specify a universal set. The calculator will automatically remove duplicates and sort the results.

5. Frequently Asked Questions (FAQ)

Q1: What format should I use for set elements?
A: Elements can be numbers, letters, or words, separated by commas (e.g., "1,2,3" or "apple,banana,orange").

Q2: Do I need to provide a universal set?
A: Only if you want to calculate complements. The universal set defines all possible elements in your context.

Q3: How are duplicates handled?
A: The calculator automatically removes duplicate elements within each set.

Q4: What's the difference between union and intersection?
A: Union combines all elements from both sets, while intersection shows only elements common to both sets.

Q5: Can I use this for more than two sets?
A: This calculator handles two sets at a time, but you can chain operations for more complex calculations.