Complement of a Set Calculator – Find A′ Online

Calculate the complement of any set with our free online complement calculator. Given a universal set U and subset A, find all elements in U that are not in A with clear set notation.

Universal Set U (comma or space separated)

Subset A (comma or space separated)

Load example:

Understanding Set Complements

The complement of a set is like taking the opposite or the "everything else." If your universal set is all students in a school, and set A is the students who play soccer, then the complement of A is everyone who doesn't play soccer. It's the flip side of the coin.

The universal set matters. The complement of {2, 4, 6} is different depending on whether your universal set is {1, 2, 3, 4, 5, 6} (complement would be {1, 3, 5}) or {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (complement would be {1, 3, 5, 7, 8, 9, 10}). Always know your universe.

Complements are fundamental to logic and probability. "Not A" in logic is the complement of A. The probability of something not happening is 1 minus the probability of it happening – that's complement thinking. In database queries, "WHERE NOT condition" finds the complement of records matching that condition.

Set Complement Notation and Formulas

A′ = {x ∈ U | x ∉ A}

Read this as: "A complement equals the set of all x in U such that x is not in A."

Alternative Notations

A′ (A prime)

A^c (A complement)

Ā (A bar)

U \ A (U minus A)

Key Properties

A ∪ A′ = U (union is universal)

A ∩ A′ = ∅ (intersection is empty)

(A′)′ = A (double complement)

|A| + |A′| = |U| (sizes add up)

Worked Examples

Example 1: Even and Odd Numbers

Find the complement of even numbers in the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Universal Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Set A (evens): {2, 4, 6, 8, 10}

Complement A′: All elements in U not in A

A′ = {1, 3, 5, 7, 9}

The complement of even numbers is the odd numbers. Together they make up all integers 1-10.

Example 2: Prime Numbers

Find the complement of prime numbers in U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Universal Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Set A (primes): {2, 3, 5, 7, 11}

Complement A′: All non-prime numbers in U

A′ = {1, 4, 6, 8, 9, 10, 12}

Note: 1 is neither prime nor composite, but it's in the complement because it's not prime.

Example 3: Vowels and Consonants

Find the complement of vowels in U = {a, b, c, d, e, f, g, h}

Universal Set U: {a, b, c, d, e, f, g, h}

Set A (vowels): {a, e}

Complement A′: All consonants in U

A′ = {b, c, d, f, g, h}

The complement of vowels gives us the consonants. If the universal set included all 26 letters, the complement would have 21 elements.

Example 4: Perfect Squares

Find the complement of perfect squares in U = {1, 2, 3, ..., 20}

Universal Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Set A (perfect squares): {1, 4, 9, 16} (since 1²=1, 2²=4, 3²=9, 4²=16)

Complement A′: All non-perfect-squares in U

A′ = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}

Out of 20 numbers, only 4 are perfect squares. The complement has 16 elements.

Example 5: Weekend and Weekdays

Find the complement of weekend days in the set of all days of the week.

Universal Set U: {Mon, Tue, Wed, Thu, Fri, Sat, Sun}

Set A (weekend): {Sat, Sun}

Complement A′: All weekdays

A′ = {Mon, Tue, Wed, Thu, Fri}

The complement of weekend days is the set of weekdays – the 5 days most people work.

Quick Fact

Georg Cantor (1845-1918), the founder of set theory, developed the concept of complements as part of his revolutionary work on infinity. Cantor showed that infinite sets can have different sizes – the set of natural numbers is "smaller" than the set of real numbers, even though both are infinite. His work was so radical that many mathematicians of his time rejected it. David Hilbert later defended Cantor, saying, "No one shall expel us from the paradise that Cantor has created." Set complements are now fundamental to mathematics, logic, and computer science.

Frequently Asked Questions

What is a universal set?

The universal set is the "everything" for your particular problem – the complete collection of elements you're considering. It defines the boundaries of your discussion. Without a universal set, the complement is undefined because "everything not in A" could mean anything in existence.

Can the complement be empty?

Yes, if A equals the universal set U, then A′ is empty. For example, if U = {1, 2, 3} and A = {1, 2, 3}, then A′ = ∅. There's nothing left outside of A because A already contains everything.

What is the complement of the empty set?

The complement of the empty set is the universal set itself. If A = ∅, then A′ = U. Nothing is in the empty set, so everything in U is "not in A." This is why ∅′ = U.

What does (A′)′ mean?

That's the complement of the complement – "the opposite of the opposite." It brings you back to the original set: (A′)′ = A. If A is "students who play soccer," A′ is "students who don't play soccer," and (A′)′ is "students who don't (not play soccer)" = "students who play soccer."

How is complement used in probability?

The probability of an event not happening is the complement of the probability of it happening: P(not A) = 1 - P(A). This is often easier to calculate. For example, the probability of rolling at least one 6 in four dice rolls is 1 - P(no 6s) = 1 - (5/6)⁴ ≈ 0.52.

What is De Morgan's Law?

De Morgan's Laws relate complements of unions and intersections: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. In words: "not (A or B)" equals "not A and not B," and "not (A and B)" equals "not A or not B." These are fundamental to logic and Boolean algebra.

Can sets contain both numbers and text?

Yes! Sets can contain any distinct elements – numbers, letters, words, objects, even other sets. This calculator handles mixed types. For example, U = {1, 2, apple, banana} is perfectly valid, and if A = {1, apple}, then A′ = {2, banana}.

Other Free Tools

Search tools