Book of Proof, third edition
Richard Hammack · Free textbook
Read §14.1, printed pp.269–270 / PDF pp.281–282, for finite matching; use §1.1 only as notation repair.
Open the sourceWhat is a number?
A quantity, a symbol, a position — or a pattern of relationships?
Prerequisites: None
Understand
Objectives
- Construct a complete matching between two finite collections.
- Distinguish cardinality from a written numeral and an ordinal position.
- Test whether rearranging objects changes their cardinality.
Start with something you can see
Put five stones beside five taps. Predict whether changing their spacing changes how many there are. Pair each stone with one tap; every object has exactly one partner and nothing is left over. The pairing survives the rearrangement. Now remove one tap: one stone has no partner. The short MIT passage starts with this matching test. [1]
Give the idea a precise name
For finite sets A and B, a bijection is a matching that gives every member of A exactly one distinct member of B and uses every member of B. They then have equal cardinality, written |A|=|B|. The written mark “5” is a numeral for a number; it is not five stones. Counting also supplies an order: fifth names a position, not the size of the whole collection.
|A| = |B|
The vertical bars mean the finite number of members; A and B name the collections. A successor rule gives another view of natural numbers: start at zero and apply “next” repeatedly. Here S(S(S(0))) names three. This course includes zero among the natural numbers. A rule or encoding is a way to represent the number, not a claim that physical objects are made of symbols.
Work one small world
Let A={red,blue,green} and B={4,7,9}. Match red→4, blue→7, green→9. This uses all three on each side, so both cardinalities are three. Replacing the label 9 by a square object does not change the count. The numeral 3 and the binary numeral 11 name that same number in different notations.
| Collection | Partners | Left over |
|---|---|---|
| 5 stones, 5 taps | 5 | 0 |
| 5 stones, 4 taps | 4 | 1 stone |
| 0 stones, 0 taps | 0 | 0 |
Counterexample to “equal cardinality means identical appearance”: {red,blue,green} and {4,7,9} have no matching labels or materials, yet the displayed bijection proves equal size. Conversely, “11” is a numeral whose value depends on its base; in binary it is three, not eleven.
THE BRIDGEThis is the first representation change: discard the material while retaining a complete matching. Later independence will similarly forget coordinates but keep a relation.
WHERE THIS IDEA STOPSEqual cardinality of finite sets is not the claim that their members resemble one another. Alternative set- or function-based number encodings are optional; matching does not require them.
Experiment
Predict the result for 5 versus 5, 5 versus 4, and 0 versus 0 before drawing pairs. Rearrange without adding or removing objects; count unmatched objects afterward.
Can two collections be equally numerous even when no object in one resembles anything in the other?
Check & explain
Which change leaves five stones with the same cardinality?
- Spread the same five stones apart
- Remove one stone
- Add a sixth stone
A={red,blue,green} and B={4,7,9}. How many pairs does a bijection use?
Teach back
Match two differently labeled collections of four objects. Explain why the match proves equal cardinality but neither the labels nor an ordinal position define that cardinality.
- Exhibit four distinct complete pairs.
- Separate written numeral from quantity.
- Explain why changing labels preserves the matching.
Read deeper
Prealgebra 2e
OpenStax · Open textbook
Use chapter 1 whole numbers, counting and notation if needed.
Open the sourceLecture 2: Cantor's Theory of Cardinality (Size)
MIT OpenCourseWare · Lecture · 85 min
Start at 2:30. Watch the short core 2:30–4:45; then the longer passage 2:30–22:30. The full 85-minute lecture is available from the same link.
Open the source