Union and Intersection of Sets: Formulas with Examples

The union of two sets A and B, written A union B, collects every element that is in A or in B (or in both). The intersection, written A intersect B, keeps only the elements that are in both A and B at the same time.

They matter because almost every "or" and "and" question in counting, probability and databases is secretly a union or intersection question. The one formula that stops double-counting is the union formula, and it appears in exams again and again.

Key Concepts

• Union A union B: {x such that x is in A or x is in B}. "Or" here includes "both".
• Intersection A intersect B: {x such that x is in A and x is in B}.
• Disjoint sets: sets with empty intersection; their union is just the two sets placed side by side.
• n(A): the number of elements of A, also called its cardinal number.

Formulas

• n(A union B) = n(A) + n(B) minus n(A intersect B). Add the two counts, then subtract the overlap once, because shared elements were counted twice.
• For three sets: n(A union B union C) = n(A) + n(B) + n(C) minus n(A intersect B) minus n(B intersect C) minus n(A intersect C) plus n(A intersect B intersect C).
• Useful identities: A union empty set = A; A intersect U = A; A union U = U; A intersect empty set = empty set.
• Limits: the counting formulas need finite sets; with infinite sets only the set identities are safe.

Worked Example

In a class of 50 students, 30 play cricket (set C), 25 play football (set F), and 12 play both. How many play at least one of the two games?

1. We need n(C union F), and we know n(C) = 30, n(F) = 25, n(C intersect F) = 12.
2. Apply the formula: 30 + 25 minus 12 = 43.
3. So 43 students play at least one game.
4. Bonus: students playing neither = 50 minus 43 = 7.

Common Misconceptions

• "Union is just adding the counts." Adding works only for disjoint sets. With any overlap you must subtract the intersection, or shared elements get counted twice.
• "Intersection of two non-empty sets cannot be empty." It can: {1, 2} intersect {3, 4} is the empty set, and such sets are called disjoint.
• "A union B contains everything twice." Sets never repeat elements; the union lists each shared element exactly once.

Real-World Applications

• Databases: an OR search is a union of result sets; an AND search is an intersection.
• Probability: P(A or B) = P(A) + P(B) minus P(A and B) is the same formula in disguise.
• Marketing: counting customers reached by campaign A, campaign B, or both, without double-counting.

Union gathers everything from both sets with no repeats; intersection keeps only what they share. Remember the subtraction in n(A union B) = n(A) + n(B) minus n(A intersect B), and double-counting errors disappear.