MathematicsChapter 3: Sets and Relations
Union Intersection Difference
From two sets you build new sets with everyday words.
Learning objectives
- Explain Union Intersection Difference in simple mathematical language.
- Apply union and intersection correctly.
- Apply difference and complement correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
From two sets you build new sets with everyday words. Union A union B is the set of elements in A or B or both. Intersection A intersect B is the set of elements in both. Difference A - B is elements in A that are not in B; it is not commutative, because B - A is a different question. Complement A' (or A^c) is U - A, everything in the universe U that is not in A. These four operations are the algebra of sets.
Where / symbols
- Union
- Union A union B is the set of elements in A or B or both.
- or
- From two sets you build new sets with everyday words.
- both
- Union A union B is the set of elements in A or B or both.
- Intersection
- Intersection A intersect B is the set of elements in both.
- Difference
- Difference A - B is elements in A that are not in B; it is not commutative, because B - A is a different question.
- A - B
- Difference A - B is elements in A that are not in B; it is not commutative, because B - A is a different question.
- not commutative
- Difference A - B is elements in A that are not in B; it is not commutative, because B - A is a different question.
- Complement
- Complement A' (or A^c) is U - A, everything in the universe U that is not in A.
- U - A
- Complement A' (or A^c) is U - A, everything in the universe U that is not in A.
- A intersect A'
- From that, A intersect A' is empty (nothing is both in and out of A) and A union A' = U (everything in U is in A or out of A).
Overview
Join and overlap first; then cut away:
Union and Intersection
Union uses or/both: if x is in at least one of the two sets, it is in the union. Intersection uses both: x must sit in A and in B. If A = 1, 2, 3 and B = 3, 4, then A union B = 1, 2, 3, 4 and A intersect B = 3. Union grows; intersection shrinks (or stays).
Examples {1, 2} union {2, 3} = {1, 2, 3} · {1, 2} intersect {2, 3} = {2} · A union A = A
Difference and Complement
A - B takes A and throws out anything that also belongs to B. It is not commutative: A - B and B - A usually differ. Complement needs a universe U first; then A' = U - A. From that, A intersect A' is empty (nothing is both in and out of A) and A union A' = U (everything in U is in A or out of A).
Examples {1, 2, 3} - {3, 4} = {1, 2} · {3, 4} - {1, 2, 3} = {4} · If U = {1, 2, 3} and A = {1}, then A' = {2, 3}
WHAT — Core idea
From two sets you build new sets with everyday words. Union A union B is the set of elements in A or B or both. Intersection A intersect B is the set of elements in both. Difference A - B is elements in A that are not in B; it is not commutative, because B - A is a different question. Complement A' (or A^c) is U - A, everything in the universe U that is not in A. These four operations are the algebra of sets.
WHY — Why it matters
A intersect A' empty and A union A' = U are the same idea as a switch: every object of U is either in A or in the complement, never both. Difference does not need U, but complement does.
HOW — Method and examples
1. UNION AND INTERSECTION: Union uses or/both: if x is in at least one of the two sets, it is in the union. 2. DIFFERENCE AND COMPLEMENT: A - B takes A and throws out anything that also belongs to B.
HOW — Answer in an exam
A - B is not B - A, so difference is not commutative. Complement is not the same as difference unless the first set is the whole universe. Always name U before you write A'. 'Or' in union includes both; it is not exclusive.
Explanation
A intersect A' empty and A union A' = U are the same idea as a switch: every object of U is either in A or in the complement, never both. Difference does not need U, but complement does.
In simple words
A - B is not B - A, so difference is not commutative. Complement is not the same as difference unless the first set is the whole universe. Always name U before you write A'. 'Or' in union includes both; it is not exclusive.
Key points
- Union
- or
- both
- Intersection
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Union Intersection Difference
Sticky Notes · Quick Revision
WHAT
- From two sets you build new sets with everyday words.
Key Points
- From two sets you build new sets with everyday words.
- Union uses or/both: if x is in at least one of the two sets, it is in the union.
- A - B takes A and throws out anything that also belongs to B.
- A intersect A' empty and A union A' = U are the same idea as a switch: every object of U is either in A or in the complement, never both.
HOW
- 1. UNION AND INTERSECTION: Union uses or/both: if x is in at least one of the two sets, it is in the union.
- 2. DIFFERENCE AND COMPLEMENT: A - B takes A and throws out anything that also belongs to B.
Examples
- {1, 2} union 2, 3 = 1, 2, 3
- 1, 2 intersect 2, 3 = 2
- A union A = A
- 1, 2, 3 - 3, 4 = 1, 2
- 3, 4 - 1, 2, 3 = 4
Common Mistake
- A - B is not B - A, so difference is not commutative.
- Complement is not the same as difference unless the first set is the whole universe.
Exam Tip
- A - B is not B - A, so difference is not commutative. Complement is not the same as difference unless the first set is the whole universe. Always name U before you write A'. 'Or' in union includes both; it is not exclusive.
60-second revision
- From two sets you build new sets with everyday words.
- Union uses or/both: if x is in at least one of the two sets, it is in the union.
- A · B takes A and throws out anything that also belongs to B.
- A intersect A' empty and A union A' = U are the same idea as a switch: every object of U is either in A or in the complement, never both.
- A · B is not B · A, so difference is not commutative.
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