MathematicsChapter 3: Sets and Relations
What Makes A Set
A set is a well-defined collection of distinct objects.
Learning objectives
- Explain What Makes A Set in simple mathematical language.
- Apply well-defined or not correctly.
- Apply distinct objects correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
A set is a well-defined collection of distinct objects. Well-defined means a clear yes/no test: you can decide, with no argument, whether an object belongs. 'Tall students in this class' is not a set, because tall has no fixed cut-off. 'Objects with mass less than 1 kg' is a set, because a balance gives a definite answer. Distinct means you do not list the same member twice. Sets matter because almost every later idea (union, relation, function) starts here.
Where / symbols
- well-defined
- A set is a well-defined collection of distinct objects.
- distinct
- A set is a well-defined collection of distinct objects.
- Tall students
- 'Tall students in this class' is not a set, because tall has no fixed cut-off.
- not a set
- 'Tall students in this class' is not a set, because tall has no fixed cut-off.
- mass
- 'Objects with mass less than 1 kg' is a set, because a balance gives a definite answer.
- 1 kg
- 'Objects with mass less than 1 kg' is a set, because a balance gives a definite answer.
- natural numbers less than 1
- The empty collection is allowed: there are no natural numbers less than 1, so that set has no members.
- even primes {2}
- A set with one member, such as the even primes {2}, is still a set.
- roster
- Order in a roster does not matter either: {2, 1} is the same set as {1, 2}.
Overview
A membership test, then the 'no repeats' rule:
Well-Defined Or Not
Ask: can two honest people disagree? If yes, the collection is not well-defined. Beautiful days, interesting books, and tall students fail. Even numbers, vowels in ENGLISH, and masses strictly less than 1 kg pass, because the rule is sharp. The objects themselves may be numbers, letters, or physical things; the test is what matters.
Examples Even integers (a set) · Mass < 1 kg (a set) · Tall students (not a set)
Distinct Objects
Members are distinct: {1, 2, 2, 3} is written {1, 2, 3}. Repeating a name does not create a new element. Order in a roster does not matter either: {2, 1} is the same set as {1, 2}. Later, ordered pairs will care about order; ordinary sets do not.
Examples {1, 2, 2} = {1, 2} · {a, b} = {b, a} · Set of even primes is {2}
WHAT — Core idea
A set is a well-defined collection of distinct objects. Well-defined means a clear yes/no test: you can decide, with no argument, whether an object belongs. 'Tall students in this class' is not a set, because tall has no fixed cut-off. 'Objects with mass less than 1 kg' is a set, because a balance gives a definite answer. Distinct means you do not list the same member twice. Sets matter because almost every later idea (union, relation, function) starts here.
WHY — Why it matters
The empty collection is allowed: there are no natural numbers less than 1, so that set has no members. A set with one member, such as the even primes {2}, is still a set. Distinct does not mean 'different kinds'; it means each listed object appears once.
HOW — Method and examples
1. WELL-DEFINED OR NOT: Ask: can two honest people disagree? 2. DISTINCT OBJECTS: Members are distinct: {1, 2, 2, 3} is written {1, 2, 3}.
HOW — Answer in an exam
Subjective words (tall, nice, famous) destroy well-defined. 'Students taller than 160 cm' would be a set, because 160 cm is a cut-off. Do not list repeats in a roster; the examiner wants distinct elements.
Explanation
The empty collection is allowed: there are no natural numbers less than 1, so that set has no members. A set with one member, such as the even primes {2}, is still a set. Distinct does not mean 'different kinds'; it means each listed object appears once.
In simple words
Subjective words (tall, nice, famous) destroy well-defined. 'Students taller than 160 cm' would be a set, because 160 cm is a cut-off. Do not list repeats in a roster; the examiner wants distinct elements.
Key points
- well-defined
- distinct
- Tall students
- not a set
- tall students
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.What Makes A Set
Sticky Notes · Quick Revision
WHAT
- A set is a well-defined collection of distinct objects.
Key Points
- A set is a well-defined collection of distinct objects.
- Ask: can two honest people disagree?
- Members are distinct: {1, 2, 2, 3} is written {1, 2, 3}.
- The empty collection is allowed: there are no natural numbers less than 1, so that set has no members.
HOW
- 1. WELL-DEFINED OR NOT: Ask: can two honest people disagree?
- 2. DISTINCT OBJECTS: Members are distinct: {1, 2, 2, 3} is written {1, 2, 3}.
Examples
- Even integers (a set)
- Mass < 1 kg (a set)
- Tall students (not a set)
- 1, 2, 2 = 1, 2
- a, b = b, a
Common Mistake
- Subjective words (tall, nice, famous) destroy well-defined.
- 'Students taller than 160 cm' would be a set, because 160 cm is a cut-off.
Exam Tip
- Subjective words (tall, nice, famous) destroy well-defined. 'Students taller than 160 cm' would be a set, because 160 cm is a cut-off. Do not list repeats in a roster; the examiner wants distinct elements.
60-second revision
- A set is a well-defined collection of distinct objects.
- Ask: can two honest people disagree?
- Members are distinct: {1, 2, 2, 3} is written {1, 2, 3}.
- The empty collection is allowed: there are no natural numbers less than 1, so that set has no members.
- Subjective words (tall, nice, famous) destroy well-defined.
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