Math101 · Independent Ontario learning library
Math101learn.math101.caSets
A rigorous foundation in sets, membership, subsets, equality, cardinality, and power sets.
Precise definition
A set is a collection determined by membership: for each object, it is meaningful to ask whether it belongs. Sets ignore order and repetition, so $\{1,2,2\}=\{2,1\}$. Two sets are equal exactly when they have the same elements. The empty set $\varnothing$ has no elements and is a subset of every set.
Notation and mathematical language
Write $x\in A$ for membership and $A\subseteq B$ when every element of $A$ lies in $B$. A proper subset $A\subsetneq B$ is unequal to $B$. The cardinality $|A|$ counts elements for finite $A$. The power set $\mathcal P(A)$ is the set of all subsets of $A$.
Conceptual picture
A set can be specified by a roster or a membership rule. A power set is a collection whose elements are themselves sets; the empty subset and the entire original set are always included. A set of size $n$ has $2^n$ subsets because each element independently has an in/out choice.
Conditions and key results
Roster notation must make membership unambiguous, and ellipses require a clear pattern. Set-builder notation $\{x\in D:P(x)\}$ includes a domain $D$ and predicate. Infinite cardinality requires more care than finite counting; a bijection, not physical size or containment alone, compares countable sets.
A reliable strategy
- Identify the universe of possible objects and state a precise membership rule.
- Remove duplicate roster entries and ignore order when comparing sets.
- For subset claims, take an arbitrary member of the proposed subset and prove it lies in the other set.
- For a finite power set, list subsets by size and confirm the total is $2^{|A|}$.
Fully worked example
Interpretation and application
Sets define solution spaces, sample spaces, categories, database results, and mathematical structures. Real categories can overlap or have uncertain membership; the mathematical set model is exact only after the classification rule is stated.
Common mistakes
Verification and reasonableness
- Prove set equality by checking both inclusions or equivalent membership conditions.
- For finite sets, compare the listed cardinality with the number of distinct members.
- For power sets, confirm both $\varnothing$ and the original set appear and that the count is $2^n$.
Practice
- How many subsets does a five-element set have?
- Is $\varnothing\subseteq\{1,2\}$?
- Are $\{1,1,2\}$ and $\{2,1\}$ equal?
Answers and brief solutions
- $2^5=32$.
- Yes; there is no element of the empty set that violates the subset condition.
- Yes. Repetition and order do not change a set.
Further deduction
Infinite sets challenge finite intuition. The natural numbers and even natural numbers have the same cardinality because $f(n)=2n$ is a bijection between them, even though the evens form a proper subset. This does not contradict finite counting; it shows that 'proper subset means smaller cardinality' is a finite theorem. To compare infinite sets, construct a bijection or prove none exists rather than relying on visual containment.
Cantor's diagonal argument proves $|A|<|\mathcal P(A)|$ for every set $A$: no function from $A$ onto its power set can include the subset of elements not contained in their own image. Power sets therefore create a strictly larger cardinality even for infinite sets.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
How many elements are in the power set of a 6-element set?
- A set of size $n$ has $2^n$ subsets.
- $2^6=64$.
End of lesson
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