Set Operations Calculator
Set operations calculator: enter finite sets (A, B, and an optional universal set U) and compute union, intersection, difference, symmetric difference, complements, cardinalities, and probability-style ratios with clear notation.
Full original guide (expanded)
Set Operations Calculator
Type in two finite sets \(A\) and \(B\) (and optionally a universal set \(U\)) and this tool will compute union, intersection, differences, sym\-metric difference, complements, cardinalities, subset relations, and probability-style ratios such as \(|A| / |U|\).
Designed for probability, discrete maths, and CS students
Handles typical textbook notation, removes duplicates automatically, and surfaces the key relationships (\(A \subseteq B\), disjointness, complements) in one glance.
Author: CalcDomain Math Team
Reviewed by: Discrete mathematics instructor
Last updated: 2025
This calculator is an educational aid. For graded work or formal proofs, always double-check the underlying reasoning and show your working.
Interactive set operations workspace
Separate elements with commas, spaces, or line breaks. Braces are optional: a, b, c and {a, b, c} are treated identically.
When disabled, inputs are converted to lower case before comparison.
Required if you want complements and probability-style ratios. Must contain all elements of A and B.
Duplicates are ignored; A is treated as a set, not a multiset.
You can enter numbers, letters, or short labels (e.g. Mon, Tue, Wed).
Example: \(A = \{1,2,3\}\), \(B = \{3,4,5\}\), \(U = \{1,2,3,4,5,6\}\) gives \(A \cup B = \{1,2,3,4,5\}\), \(A \cap B = \{3\}\), \(A^c = \{4,5,6\}\), etc.
Basic set notation and operations
In set theory, a set is a collection of distinct elements. We usually denote sets with capital letters such as \(A, B, U\) and list elements inside braces, for example \(A = \{1,2,3\}\), \(B = \{3,4,5\}\).
Union, intersection, and difference
Given sets \(A\) and \(B\) inside a universal set \(U\):
- The union \(A \cup B\) contains every element that is in \(A\), in \(B\), or in both.
- The intersection \(A \cap B\) contains elements that are in both \(A\) and \(B\).
- The difference \(A \setminus B\) contains elements that are in \(A\) but not in \(B\).
- The symmetric difference \(A \Delta B\) contains elements that are in exactly one of \(A\) or \(B\), but not in both.
Complements and cardinalities
A universal set \(U\) is a set that contains all elements under consideration. If \(A \subseteq U\), the complement of \(A\) in \(U\) is:
The cardinality of a finite set \(A\), written \(|A|\), is simply the number of distinct elements in \(A\). When \(U\) is finite and all events of interest are subsets of \(U\), this blends naturally with basic probability:
Worked example
Let the universal set be \(U = \{1,2,3,4,5,6\}\), and define \(A = \{1,2,3\}\) and \(B = \{3,4,5\}\).
- \(A \cup B = \{1,2,3,4,5\}\)
- \(A \cap B = \{3\}\)
- \(A \setminus B = \{1,2\}\), \(B \setminus A = \{4,5\}\)
- \(A \Delta B = \{1,2,4,5\}\)
- \(A^{c} = U \setminus A = \{4,5,6\}\), \(B^{c} = \{1,2,6\}\)
- \(|A| = 3\), \(|B| = 3\), \(|U| = 6\), so \(P(A) = \frac{3}{6} = 0.5\), \(P(B) = 0.5\), \(P(A \cap B) = \frac{1}{6} \approx 0.167\).
FAQ: set operations calculator
What happens if I repeat an element?
Sets, by definition, do not track multiplicity. If you enter 1, 1, 2, 2, 2, 3, the calculator treats the set as \(\{1,2,3\}\). This is intentional and matches standard set theory.
Can I use words or symbols as elements?
Yes. Elements are handled as text tokens. For example, Mon, Tue, Wed or apple, banana, cherry are perfectly valid sets. Just avoid commas inside a single element, as commas are used to separate elements.
Why are complements and probabilities sometimes disabled?
Complements \(A^{c}\) and \(B^{c}\) are only meaningful once you specify a universal set \(U\). The calculator requires that all elements of \(A\) and \(B\) belong to \(U\). If it detects elements outside \(U\), it will show a warning and skip complements and probability-style ratios.
How can I use this tool to debug exam or homework questions?
One good workflow is: translate the question into explicit sets \(A, B, U\); type them into the calculator; note the numeric outputs \(|A|, |B|, |A \cap B|\), etc.; and then reconstruct your own step-by-step argument, using the tool as a consistency check rather than as a black box.
Formula (LaTeX) + variables + units
A \cup B = \{x : x \in A \text{ or } x \in B\}
A \cap B = \{x : x \in A \text{ and } x \in B\}
A \setminus B = \{x : x \in A \text{ and } x \notin B\}
A \Delta B = (A \setminus B) \cup (B \setminus A)
A^{c} = U \setminus A = \{x : x \in U \text{ and } x \notin A\}.
P(A) = \frac{|A|}{|U|}, \quad P(B) = \frac{|B|}{|U|}, \quad P(A \cap B) = \frac{|A \cap B|}{|U|}.
\[ A \cup B = \{x : x \in A \text{ or } x \in B\} \] \[ A \cap B = \{x : x \in A \text{ and } x \in B\} \] \[ A \setminus B = \{x : x \in A \text{ and } x \notin B\} \] \[ A \Delta B = (A \setminus B) \cup (B \setminus A) \]
\[ A^{c} = U \setminus A = \{x : x \in U \text{ and } x \notin A\}. \]
\[ P(A) = \frac{|A|}{|U|}, \quad P(B) = \frac{|B|}{|U|}, \quad P(A \cap B) = \frac{|A \cap B|}{|U|}. \]
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Last code update: 2026-01-19
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