Union Formula

The Formula

A \cup B = \{x : x \in A \text{ or } x \in B\}

When to use: Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

Quick Example

A = \{1, 2, 3\}, B = \{3, 4, 5\}. Then A \cup B = \{1, 2, 3, 4, 5\} — the 3 appears once.

Notation

A \cup B

What This Formula Means

The union of sets A and B is the set of all elements that belong to A, to B, or to both, written A \cup B.

Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

Formal View

A \cup B = \{x : x \in A \lor x \in B\}

Worked Examples

Example 1

easy
Let A = \{1, 3, 5\} and B = \{2, 3, 4\}. Find A \cup B.

Solution

  1. 1
    Recall the definition: A \cup B = \{x : x \in A \text{ or } x \in B\}. The word 'or' is inclusive — an element belongs to the union if it appears in at least one of the sets.
  2. 2
    List all elements from A = \{1,3,5\} and B = \{2,3,4\}, including each at most once: 1 (from A), 2 (from B), 3 (in both), 4 (from B), 5 (from A).
  3. 3
    Therefore A \cup B = \{1,2,3,4,5\}. Notice |A \cup B| = 5 = |A| + |B| - |A \cap B| = 3 + 3 - 1, confirming the inclusion-exclusion principle.

Answer

A \cup B = \{1, 2, 3, 4, 5\}
The union operator collects all elements from both sets. Duplicate elements are listed only once because sets contain distinct elements.

Example 2

medium
Let A = \{x \in \mathbb{R} : x > 2\} and B = \{x \in \mathbb{R} : x < 5\}. Express A \cup B in interval notation.

Common Mistakes

  • Confusing union (\cup) with intersection (\cap) — union includes ALL elements from both sets
  • Including duplicate elements in the result — sets never have duplicates, so \{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}, not \{1, 2, 2, 3\}
  • Forgetting that A \cup \emptyset = A, not \emptyset

Why This Formula Matters

Union combines overlapping categories without double-counting, a tool used in probability, data analysis, and every branch of math.

Frequently Asked Questions

What is the Union formula?

The union of sets A and B is the set of all elements that belong to A, to B, or to both, written A \cup B.

How do you use the Union formula?

Pour both sets into one container and remove duplicates. Everything from either pile ends up in the union — this is the OR operation for sets.

What do the symbols mean in the Union formula?

A \cup B

Why is the Union formula important in Math?

Union combines overlapping categories without double-counting, a tool used in probability, data analysis, and every branch of math.

What do students get wrong about Union?

Union doesn't duplicate—element 2 appears once in the result.

What should I learn before the Union formula?

Before studying the Union formula, you should understand: set.

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