Chapter 2.5 — Exercise 2.4 — Finite and Infinite Sets

Finite Sets and Infinite Sets

A set is finite if it has both a clear starting point and a clear ending point — every element can, in principle, be listed and the listing eventually stops. A set is infinite if no such ending point exists.

The empty set is a special case worth remembering: it counts as finite, since zero is itself a definite, countable number of elements — there's no ambiguity about where the listing "ends" when it never starts.

A useful habit when a set is described by a condition rather than listed directly: try to picture what listing its elements one at a time would actually look like. For "whole numbers between 5 and 45," you can imagine writing 6, 7, 8, … and eventually reaching 44, then stopping — a finite process. For "all multiples of 15," the same imagined listing — 15, 30, 45, 60, … — never has a natural stopping point, however far you continue, which is the real test for infinite.

Cardinal Number: Counting a Finite Set

The number of elements in a finite set is called its cardinal number, written n(A) for a set A.

• R = {a, b, c, d} → n(R) = 4
• The vowels in the English alphabet → n(A) = 5
• The letters of the word "MAY" → n(M) = 3
• S = {1, 2, 3, …, 98, 99} → n(S) = 99
• The empty set → n(φ) = 0

Cardinal number is only ever defined for finite sets — asking "how many elements does the set of all natural numbers have?" doesn't have a numeric answer the way n(A) = 4 does, because the listing never finishes long enough to produce a final count. This is also where the subset-counting formula from earlier exercises connects back: a set with cardinal number n(A) = n has exactly 2ⁿ subsets, so once you know a set's cardinal number, you immediately know how many subsets it has too, without listing a single one.

Spotting an Empty Set

Deciding whether a set is empty means checking whether even one element genuinely satisfies its defining condition — not assuming from how the description sounds.

• Integers strictly between 2 and 3 — empty; no integer exists in that gap.
• Natural numbers smaller than 1 — empty; natural numbers start at 1.
• Odd numbers leaving remainder 0 when divided by 2 — empty; by definition, odd numbers always leave remainder 1.
• x satisfying both x² = 4 and 3x = 9 — empty; x² = 4 forces x = ±2, but 3x = 9 forces x = 3 — no value satisfies both at once.
• Triangles whose three angles sum to less than 180° — empty; every triangle's angles sum to exactly 180°, without exception.

Finite or Infinite? A Quick Round

Classifying a set correctly sometimes means looking past its description to what it's actually counting:

A set described by a condition, like "days of the week," is just as finite as one written out in full — what matters is whether the listing genuinely stops, not whether the description happens to use a formula or a rule. {1², 2², 3², …} is the one genuine trap in this list: the ellipsis after only three terms might tempt a quick reader into assuming a short, finite pattern, but the "…" here signals the sequence keeps going indefinitely, exactly the same as it would for the natural numbers themselves.

Four Real-World Membership Checks

The exercise closes with quick judgment calls about everyday collections:

• The set of lines passing through a given point — non-empty (infinitely many lines pass through any single point), and infinite.
• Odd natural numbers divisible by 2 — empty, since no odd number is ever divisible by 2.
• {x : x is a natural number, x < 5 and x > 7} — empty; no number is simultaneously below 5 and above 7.
• A point common to two parallel lines — empty, since parallel lines by definition never meet.
• Even prime numbers — non-empty: {2} alone satisfies this, since 2 is the only even number that's also prime.

The even-primes example is worth lingering on, because it looks at first like it should be a large set — most people's instinct is that "even numbers" and "prime numbers" should overlap substantially. In fact every even number greater than 2 is divisible by 2 and therefore automatically composite, which is exactly why 2 is not just an even prime but the only one. A single-element set like {2} is still finite and still non-empty — cardinal number 1 is a perfectly ordinary answer, just an easy one to underestimate at a glance.

Six More Classifications

Rounding out the exercise, six further sets confirm the finite/infinite distinction across genuinely different kinds of collections — numeric ranges, geometric families, and calendar facts alike:

The last three rows share a common thread worth naming directly: each describes a whole family of geometric objects — lines, multiples, circles — rather than a fixed handful of specific ones. Whenever a description allows "any value of a continuous or unbounded parameter" (any height for a horizontal line, any multiplier for 5, any radius for a circle through one point), the resulting set is infinite almost by construction, without needing to check individual cases one at a time.

The Chapter, Tied Together

This exercise closes the loop on everything Chapter 2 built: a set first has to be well-defined (the chapter introduction), can be written in roster or set-builder form (Exercise 2.1), combined with others through union, intersection, and difference (Exercise 2.2), and compared for equality (Exercise 2.3) — and now, finally, classified by whether counting its elements ever finishes. Set notation reappears throughout the rest of Class 10 Mathematics, starting with Chapter 3's Polynomials, where the solution set of an equation is written and reasoned about in exactly this language.