Use bijections for finite counting and distinguish countable from uncountable sets.
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Try
A finite set of four objects has the same size as any other four-object set because we can pair their elements with no leftovers. A bijection makes that idea precise: it is a function that is both one-to-one and onto. Counting by a bijection can avoid enumerating every object.
Predict
Predict whether pairing still makes sense for infinite sets. Let N₀ be the nonnegative integers. The map n↦2n pairs N₀ with its even members, despite the latter being a proper subset. Finite intuition about a proper subset being strictly smaller does not extend unchanged to infinite cardinality.
Experiment
Write the first six pairs of n↦2n+1, then write its inverse on the positive odd integers. Explain why a finite table alone is insufficient, and why the inverse formula (m−1)/2 proves every permitted output has exactly one input.
Understand
Worked finite bijection: a subset of an n-element labeled set corresponds to a binary string of length n, with 1 meaning membership. Reading the 1 positions recovers the subset, so the correspondence is invertible. There are therefore 2ⁿ subsets, including the empty subset.
A set is countably infinite when it admits a bijection with N₀. Integers can be listed as 0,1,−1,2,−2,… . Rational numbers are countable: arrange numerator and positive denominator pairs by increasing |numerator|+denominator, reducing fractions and skipping duplicates. Every rational appears after finitely many earlier pairs. Infinite does not mean uncountable.
Cantor’s diagonal argument constructs a missing subset from any proposed list of subsets: include index n exactly when it is absent from the nth listed subset. The constructed subset differs from each listed subset at its own index.
Apply
Give a map and prove injectivity and surjectivity separately, or exhibit a two-sided inverse. A few displayed pairs illustrate a construction but do not establish that no infinite tail is missed. State whether your domain begins at 0 or 1.
Continue
Depth: cardinality compares existence of bijections, not density or numerical magnitude. The rational numbers are dense in the real line yet countable. The reals are uncountable; proving that claim requires a diagonal or equivalent argument, with care about nonunique positional expansions.
Apply · explain · revise
Practice and reasoning
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How many subsets does a set with 5 distinct elements have?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For f:N₀→positive odd integers, f(n)=2n+1, what is f(17)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which is true of an infinite set?
Which set is uncountable?
Construct a bijection from N₀={0,1,2,...} onto Z and justify that it is both injective and surjective.
Structure your argument and preserve revisions
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Prove that no function f:S→P(S) can be surjective by considering D={s∈S:s∉f(s)}.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
