1.5 Sets, Relations, and Representations
Key Takeaways
- Set operations depend on the universal set: union means in either set, intersection means in both, and complement means outside the named set but inside the universe.
- Inclusion-exclusion prevents double-counting overlap when adding set sizes.
- A relation is any set of ordered pairs, while a function is a relation in which each input has exactly one output.
- Translating among verbal, symbolic, tabular, graphic, and diagram forms is itself a tested reasoning skill.
Representations as mathematical evidence
The AEPA Domain I profile includes translating among graphic, verbal, and symbolic representations and recognizing connections between mathematical concepts. The broader Mathematics profile also includes set theory in the statistics, probability, and discrete mathematics domain. For this chapter, the key is to use sets, relations, and representations as reasoning tools rather than as isolated notation.
A set is a collection of elements. Use braces for roster notation, such as {2, 4, 6}, and set-builder notation when a rule is clearer, such as {x | x is an even integer greater than 0}. The universal set is the collection under discussion. A complement only makes sense relative to that universe. If the universe is students in one class, the complement of students taking calculus means classmates not taking calculus, not every non-calculus student in the world.
Set operations
| Operation | Meaning | Example interpretation |
|---|---|---|
| A union B | elements in A or B or both | students in algebra, geometry, or both |
| A intersection B | elements in both A and B | students taking both algebra and geometry |
| A complement | elements in the universe not in A | students in the class not taking algebra |
| A subset B | every element of A is in B | every square is a rectangle |
| Empty set | no elements | students with a score above 100 on a 100-point test |
The key counting formula is inclusion-exclusion: n(A union B) = n(A) + n(B) - n(A intersection B). The subtraction removes the overlap that was counted twice. In probability, the same structure appears as P(A union B) = P(A) + P(B) - P(A intersection B).
Worked example: Venn diagram counting
In a group of 50 candidates, 32 practiced algebra, 25 practiced geometry, and 12 practiced both. The number who practiced algebra or geometry is 32 + 25 - 12 = 45. Therefore 50 - 45 = 5 practiced neither. A common wrong answer is 7, from subtracting 25 from 32, but that comparison ignores the overlap and the universal set.
Relations and functions
A relation is any set of ordered pairs. The domain is the set of inputs, and the range is the set of outputs. A function is a relation that assigns each input exactly one output. The relation {(1, 2), (2, 4), (3, 6)} is a function. The relation {(1, 2), (1, 3), (2, 4)} is not a function because input 1 has two outputs.
Relations can also be described by properties. A relation is reflexive on a set if every element is related to itself. It is symmetric if a related to b implies b related to a. It is transitive if a related to b and b related to c imply a related to c. A relation that has all three properties is an equivalence relation. For example, "has the same parity as" on the integers is reflexive, symmetric, and transitive; it partitions integers into even and odd equivalence classes.
Translating among representations
A verbal statement might say, "The cost is 12 dollars plus 3 dollars per student." Symbolically, C = 12 + 3s. In a table, each increase of 1 in s raises C by 3. On a graph, the y-intercept is 12 and the slope is 3 dollars per student. Each representation emphasizes a different feature, and the exam may ask which one best supports a conclusion.
When translating, preserve restrictions. If x is the number of students, the domain is usually whole numbers at least 0, even though the line C = 12 + 3x can be drawn for all real x. If a graph shows only a closed interval, the domain and range come from the displayed interval, not from the formula you might imagine outside it.
Common traps
Do not confuse union with intersection; union is broad and intersection is shared. Do not list repeated elements in a set as if multiplicity matters; ordinary sets contain each element once. Do not identify the range before checking all outputs. Do not call every relation a function. Do not ignore open and closed endpoints on a graph, because they determine whether boundary values are included.
For AEPA preparation, practice moving in both directions: create a table from a formula, write a formula from a verbal rule, sketch a graph from a table, and describe a graph in words. The goal is not artistic graphing; it is preserving the mathematical relationships across formats.
Representation checklist
Before choosing a representation, ask what must be preserved. A table preserves individual input-output pairs and is useful for finite patterns. A graph preserves shape, intercepts, intervals, and approximate behavior. A formula preserves exact symbolic relationships and supports algebraic manipulation. A Venn diagram preserves overlap and complements. A verbal description preserves context and units. Good mathematical communication chooses the form that makes the intended relationship easiest to verify.
Also watch the difference between a displayed model and a contextual model. A line may extend forever as an algebraic object, but a word problem about ticket sales may restrict inputs to nonnegative whole numbers. A probability Venn diagram may use decimals, fractions, or counts, but the universal set must stay fixed. A relation shown by arrows may be a function from left to right but not have an inverse function from right to left. These distinctions are exactly where representation items become reasoning items rather than notation items.
When reviewing missed problems, write the same relationship in two alternate forms. If you cannot translate it, the original representation was probably memorized rather than understood.
In a group of 50 candidates, 32 studied algebra, 25 studied geometry, and 12 studied both. How many studied neither algebra nor geometry?
Which description explains why {(1, 2), (2, 4), (1, 3)} is not a function?
Which three properties must a relation have to be an equivalence relation?