Chapter 2.4 — Exercise 2.3 — Equal Sets

The Definition of Equal Sets

Sets A and B are equal if every element of A is in B, and every element of B is in A — written A = B. That's exactly the two-way subset condition from the previous exercise, applied in both directions at once.

Order never matters, and neither does how many times an element is mentioned in the original description — only the final collection of distinct elements counts.

It's worth being precise about what equal sets are not. Two sets can share several elements, or even have the same number of elements, without being equal — {1, 2, 3} and {2, 3, 4} both have three elements and two of them in common, but they're still different sets, because 1 belongs to the first and not the second. Genuine equality demands that every single element match in both directions, with no exceptions on either side — sharing "most" elements, or even "all but one," is never enough.

Three Words, One Set of Letters

A clean illustration: A is the set of letters in "FOLLOW," B is the set of letters in "FLOW," and C is the set of letters in "WOLF." Written in roster form:

Every element of A is in B, and every element of B is in A, so A = B — and the same check confirms B = C and A = C too. All three sets, despite coming from visibly different words, contain exactly the same four letters. FOLLOW's repeated O and L simply collapse to single elements once it becomes a set, which is exactly why a seven-letter word can still produce a four-element set.

Seven Pairs, Checked at a Glance

With A = {1, 2, 3}, B = {the first three natural numbers}, C = {a, b, c, d}, D = {d, c, a, b}, E = {a, e, i, o, u}, and F = {vowels in the English alphabet}, each pair is either identical in content or isn't:

Deciding Equality From Descriptions

When one side of a comparison is written in set-builder form, the safest first move is always to convert it to roster form before comparing anything.

• {a, b, c, d} and {d, c, a, b} — both list exactly the same four letters, so A = B.
• {4, 8, 12, 16} and {8, 4, 16, 18} — the second set has 18 where the first has 12, so A ≠ B despite three shared elements.
• {2, 4, 6, 8, 10} and {x : x is a positive even integer, x < 10} — the second set in roster form is {2, 4, 6, 8}, missing 10, so A ≠ B.
• {x : x is a multiple of 10} and {10, 15, 20, 25, 30, …} — the first is {10, 20, 30, …}, but the second also includes 15 and 25, which aren't multiples of 10, so A ≠ B.

Explaining an Inequality, Not Just Stating It

A further set of questions asks you to justify why two sets are not equal — which means naming a specific element that appears in one but not the other.

• {1, 2, …, 10} ≠ {x : x ∈ N, 1 < x < 10} — the right side excludes both endpoints, giving {2, 3, …, 9}; 1 and 10 are in the left set but not the right.
• {2, 4, 6, 8, 10} ≠ {x : x = 2n+1, n ∈ N} — the right side is odd numbers {3, 5, 7, …}, sharing nothing with the left set at all.
• {5, 15, 30, 45} ≠ {x : x is a multiple of 15} — the full multiples-of-15 set is {15, 30, 45, 60, …}; 5 isn't a multiple of 15, so it doesn't belong on the right.
• {2, 3, 5, 7, 9} ≠ {x : x is a prime number} — 9 is in the left set but isn't prime (9 = 3×3), and 11 is prime but missing from the left set.

Listing Every Subset

A subset is any collection whose elements all come from within a given set, including both the empty set and the set itself. Listing every subset of a set with n elements always produces exactly 2ⁿ of them — the same formula from Exercise 2.2, applied here to five different sets, ranging from just 2 elements up to 4.

The reason the count is always a power of 2 is worth seeing directly: building a subset means deciding, one element at a time, whether that element is "in" or "out" of the subset you're forming. Each of the n elements gets exactly 2 independent choices, so the total number of possible subsets is 2 multiplied by itself n times over — 2ⁿ. For a set with 4 elements, that's 2×2×2×2 = 16 distinct in/out combinations, matching sets D and E below exactly.

Sets D and E both have 4 elements, so both have exactly 16 subsets each — the count depends only on how many elements a set has, never on what those elements actually are. D contains letters and E contains perfect squares, yet their subset counts are identical, which is a useful sanity check whenever two very different-looking sets happen to be the same size.

Where Equal Sets Reappear

The equality test from this exercise — A ⊆ B and B ⊆ A together — is what later chapters lean on whenever two differently-described solution sets need to be shown identical. The subset-counting formula, 2ⁿ, carries forward unchanged into Exercise 2.4, which asks a related but different question: not how many elements a set has, but whether it's even possible to finish counting them. For the union/intersection groundwork behind subsets, see Exercise 2.2.