Sets

The One Thing

Sets are well-defined collections of distinct objects whose relationships can be analysed through membership, subset relations, operations, algebraic laws, and Venn diagrams. The central counting technique is to account for overlaps correctly, especially by subtracting common elements when using union formulas.

Definitions and Results

• Set: A well-defined collection of distinct objects, written inside braces; for example, $A=\{1,2,3\}$. Membership must be unambiguous.
• Element or Member: An object belonging to a set. $x\in A$ means that $x$ is an element of $A$, whereas $x\notin A$ means that it is not.
• Roster Form: A representation of a set by listing all its elements within braces.
• Set-Builder Form: A representation that states the common property of the elements; for example, $\{x:x\text{ is a natural number less than }5\}$.
• Descriptive Form: A representation using words to describe the elements of a set.
• Finite Set: A set containing a limited number of elements.
• Infinite Set: A set containing endlessly many elements.
• Empty Set: A set containing no elements, denoted by $\varnothing$ or $\{\}$.
• Singleton Set: A set containing exactly one element.
• Universal Set: The set containing all objects under consideration in a particular discussion, usually denoted by $U$.
• Subset: $A\subseteq B$ means that every element of $A$ is also an element of $B$.
• Proper Subset: $A\subset B$ means that $A\subseteq B$ but $A\ne B$.
• Power Set: The set of all subsets of $A$, denoted by $P(A)$. If $A$ has $n$ elements, then $P(A)$ has $2^n$ elements, and $A$ has $2^n-1$ proper subsets.
• Union: $A\cup B$ contains all elements belonging to $A$, to $B$, or to both.
• Intersection: $A\cap B$ contains only elements common to both $A$ and $B$.
• Disjoint Sets: $A$ and $B$ are disjoint if they have no common element; therefore, $A\cap B=\varnothing$.
• Difference of Sets: $A-B$ contains elements belonging to $A$ but not to $B$.
• Complement: The complement of $A$, written $A^{\prime }$ or $A^c$, contains elements of $U$ that are not in $A$:

$$A^{\prime }=U-A.$$

• Venn Diagram: A graphical representation of sets using closed curves, usually circles, inside a rectangle representing the universal set.
• Cardinality: The number of elements in a set, denoted by $n(A)$ or $|A|$.
• Ordered Pair and Cartesian Product: $A\times B$ consists of ordered pairs $(a,b)$, where $a\in A$ and $b\in B$. For finite sets,

$$n(A\times B)=n(A)n(B).$$

Important structural results include the following:

• The order of elements does not matter, and repeated elements are written only once.
• Two sets are equal if and only if they contain exactly the same elements.
• For every set $A$, $\varnothing \subseteq A$ and $A\subseteq A$.
• If $A\subseteq B$, then

$$A\cap B=A,A\cup B=B.$$

• Union is commutative and associative:

$$A\cup B=B\cup A,$$ $$(A\cup B)\cup C=A\cup (B\cup C).$$

• Intersection is commutative and associative:

$$A\cap B=B\cap A,$$ $$(A\cap B)\cap C=A\cap (B\cap C).$$

• Distributive laws:

$$A\cap (B\cup C)=(A\cap B)\cup (A\cap C),$$ $$A\cup (B\cap C)=(A\cup B)\cap (A\cup C).$$

• Identity laws:

$$A\cup \varnothing =A,A\cap U=A,$$ $$A\cup U=U,A\cap \varnothing =\varnothing .$$

• Complement laws:

$$A\cup A^{\prime }=U,A\cap A^{\prime }=\varnothing ,$$ $$U^{\prime }=\varnothing ,{\varnothing }^{\prime }=U.$$

• Involution law:

$$(A^{\prime }{)}^{\prime }=A.$$

• De Morgan’s laws:

$$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },$$ $$(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }.$$

• Difference may be expressed using complement:

$$A-B=A\cap B^{\prime }.$$

• For finite sets:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

• If $A$ and $B$ are disjoint finite sets:

$$n(A\cup B)=n(A)+n(B).$$

• For three finite sets:

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C).$$

• In a Venn diagram, the rectangle represents $U$, while enclosed regions represent individual sets and their overlaps.

Worked Methods

1. Representing a Set

• Identify the objects that belong to the collection.
• Confirm that membership is unambiguous.
• Use one of the three standard representations:

- Roster form: list the elements in braces, such as $$A=\{1,2,3\}.$$ - Set-builder form: state a defining property, such as $$\{x:x\text{ is a natural number less than }5\}.$$ - Descriptive form: describe the elements in words.

• Write repeated elements only once, since sets contain distinct objects and the order of elements does not matter.

2. Testing Equality, Subsets, and Proper Subsets

• To test whether $A=B$, compare their elements in both directions.
• The sets are equal if and only if they contain exactly the same elements.
• To test whether $A\subseteq B$, check that every element of $A$ belongs to $B$.
• To test whether $A\subset B$, verify both $A\subseteq B$ and $A\ne B$.
• Remember the general results:

$$\varnothing \subseteq A,A\subseteq A.$$

3. Finding a Power Set

• List the empty subset $\varnothing$.
• List all one-element subsets.
• List all two-element subsets, and continue systematically.
• Include the original set as a subset.
• If the original set has $n$ elements, check that the final number of subsets is

$$2^n.$$

• The number of proper subsets is

$$2^n-1.$$

4. Performing Set Operations

• For $A\cup B$, collect every element in $A$, $B$, or both, writing each element once.
• For $A\cap B$, retain only elements common to both sets.
• For $A-B$, retain elements in $A$ that are not in $B$.
• For $A^{\prime }$ or $A^c$, identify the universal set $U$ and remove the elements of $A$:

$$A^{\prime }=U-A.$$

• For disjoint sets, verify:

$$A\cap B=\varnothing .$$

5. Using Venn Diagrams

• Draw a rectangle to represent the universal set $U$.
• Draw closed curves, usually circles, for the individual sets.
• Place common elements in overlapping regions.
• Interpret regions according to the operation:

- union represents membership in $A$, $B$, or both; - intersection represents common membership; - difference represents membership in one set but not the other; - complement represents the part of $U$ outside the relevant set.

• Use the diagram to identify overlaps and avoid double-counting.

6. Counting Two and Three Sets

For two finite sets:

• Find $n(A)$.
• Find $n(B)$.
• Find the overlap $n(A\cap B)$.
• Add $n(A)$ and $n(B)$.
• Subtract the overlap once:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

• If the sets are disjoint, the overlap is empty, so:

$$n(A\cup B)=n(A)+n(B).$$

For three finite sets:

• Add the cardinalities of the three sets.
• Subtract each pairwise intersection:

$$n(A\cap B),n(B\cap C),n(C\cap A).$$

• Add the triple intersection once:

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C).$$

• This corrects for elements counted more than once.

7. Rewriting and Applying Set Laws

• Translate the operation into its logical meaning: union corresponds to “or,” intersection to “and,” and complement to “not.”
• Apply commutative, associative, distributive, identity, complement, involution, or De Morgan’s laws as appropriate.
• For a difference, use:

$$A-B=A\cap B^{\prime }.$$

• Check the result through element membership or a Venn diagram when necessary.

Where It Goes Wrong

• Treating a vague collection as a set, even though membership is not unambiguous.
• Treating repeated elements as separate elements or assuming that the order of elements changes a set.
• Confusing $A\subseteq B$ with $A\subset B$: a proper subset must not be equal to the larger set.
• Forgetting that the complement is defined relative to the universal set $U$, so $A^{\prime }=U-A$.
• Confusing union, which includes elements in either set or both, with intersection, which includes only common elements.
• Adding overlapping cardinalities without subtracting the common elements, causing double-counting; for three sets, also omitting the final addition of $n(A\cap B\cap C)$.

What Gets Asked

This material supports questions requiring students to:

• Define a set, element, finite set, infinite set, empty set, singleton set, universal set, subset, proper subset, power set, union, intersection, difference, complement, Venn diagram, cardinality, and Cartesian product.
• Represent a collection in roster form, set-builder form, and descriptive form.
• Determine whether a collection is well-defined.
• Identify whether an object belongs to a set using $\in$ and $\notin$.
• Test equality, subset, proper subset, and disjointness using element membership.
• List the power set and calculate the number of subsets and proper subsets.
• Calculate unions, intersections, differences, and complements.
• Apply the commutative, associative, distributive, identity, complement, involution, and De Morgan’s laws.
• Rewrite a difference using $A-B=A\cap B^{\prime }$.
• Interpret or complete Venn diagrams, including the universal set, individual regions, overlaps, differences, and complements.
• Calculate the cardinality of unions of two or three finite sets.
• Apply the disjoint-set formula.
• Determine the number of ordered pairs in $A\times B$ using

$$n(A\times B)=n(A)n(B).$$