Exercise 2.1 answers: Relations and Functions

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1. Two ordered pairs are equal only when their first parts match and their second parts match.
2. Match the first parts: x/3 + 1 = 5/3.
3. Subtract 1 from both sides: x/3 = 5/3 − 1 = 2/3, so x = 2.
4. Match the second parts: y − 2/3 = 1/3.
5. Add 2/3 to both sides: y = 1/3 + 2/3 = 1.

Answerx = 2, y = 1

Watch this explained “One equation becomes two”, 7:30 into Why writing a pair in order carries information a set cannot

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1. A has 3 elements, so n(A) = 3.
2. B = {3, 4, 5} has 3 elements, so n(B) = 3.
3. For any two finite sets, n(A × B) = n(A) × n(B).
4. So n(A × B) = 3 × 3 = 9.

Answer9

Watch this explained “Where the product comes from”, 0:39 into Counting all the pairs, and why swapping the two sets gives something else

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1. G × H pairs every element of G with every element of H, first from G, then from H.
2. G × H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}.
3. H × G pairs every element of H with every element of G, first from H.
4. H × G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}.
5. The two sets have the same number of pairs, but they are not the same set, because the order inside each pair is different.

AnswerG × H = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}; H × G = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}

Watch this explained “Three against one”, 2:37 into Counting all the pairs, and why swapping the two sets gives something else

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(i) P × Q = {(m, n),(n, m)}

1. P = {m, n} and Q = {n, m} are actually the same set, {m, n}.
2. P × Q must contain every pair with the first entry from P and second entry from Q: (m, m), (m, n), (n, m), (n, n).
3. The given statement lists only 2 of these 4 pairs, so it is missing (m, m) and (n, n).

AnswerFalse. Correct statement: P × Q = {(m, m), (m, n), (n, m), (n, n)}.

(ii) A × B is a non-empty set of ordered pairs

1. By definition, A × B is exactly the set of ordered pairs (x, y) with x ∈ A and y ∈ B.
2. If A and B are both non-empty, at least one such pair exists, so A × B is non-empty.

AnswerTrue

(iii) A × (B ∩ φ) = φ

1. B ∩ φ means the elements common to B and the empty set.
2. The empty set has no elements, so B ∩ φ = φ, whatever B is.
3. Crossing any set with the empty set gives the empty set, so A × (B ∩ φ) = A × φ = φ.

AnswerTrue

Watch this explained “The same set on both sides”, 6:44 into Why writing a pair in order carries information a set cannot

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1. A × A × A contains every ordered triplet (a, b, c) where a, b, c are each chosen from A.
2. A has 2 elements, so there are 2 × 2 × 2 = 8 triplets.
3. List them: (−1,−1,−1), (−1,−1,1), (−1,1,−1), (−1,1,1), (1,−1,−1), (1,−1,1), (1,1,−1), (1,1,1).

AnswerA × A × A = {(−1,−1,−1), (−1,−1,1), (−1,1,−1), (−1,1,1), (1,−1,−1), (1,−1,1), (1,1,−1), (1,1,1)}

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1. A is the set of all first entries appearing in the pairs: a and b.
2. B is the set of all second entries appearing in the pairs: x and y.
3. Check: crossing A = {a, b} with B = {x, y} gives exactly the 4 given pairs.

AnswerA = {a, b}, B = {x, y}

Watch this explained “Reading a product backwards”, 10:55 into Counting all the pairs, and why swapping the two sets gives something else

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(i) A × (B ∩ C) = … (A × C)

1. B ∩ C = {1,2,3,4} ∩ {5,6}. These two sets share no elements, so B ∩ C = φ.
2. So A × (B ∩ C) = A × φ = φ (the empty set).
3. A × B has second entries from {1,2,3,4} and A × C has second entries from {5,6}; they never match, so (A × B) ∩ (A × C) = φ too.
4. Both sides equal φ, so the statement is verified true.

AnswerYes, A × (B ∩ C) = (A × B) ∩ (A × C); both sides equal φ.

(ii) A × C is a subset of B × D

1. A × C = {(1,5), (1,6), (2,5), (2,6)}.
2. Every first entry of A × C (1 or 2) is in B, and every second entry (5 or 6) is in D.
3. So every pair of A × C is also a pair of B × D, which means A × C ⊆ B × D.

AnswerYes, A × C is a subset of B × D.

Watch this explained “Across what they share”, 8:11 into Counting all the pairs, and why swapping the two sets gives something else

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1. A × B = {(1,3), (1,4), (2,3), (2,4)}, which has 4 elements.
2. A set with n elements has 2ⁿ subsets, so A × B has 2⁴ = 16 subsets.
3. List them, from the empty set up to the whole set: {}, {(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}, {(1,3),(1,4)}, {(1,3),(2,3)}, {(1,3),(2,4)}, {(1,4),(2,3)}, {(1,4),(2,4)}, {(2,3),(2,4)}, {(1,3),(1,4),(2,3)}, {(1,3),(1,4),(2,4)}, {(1,3),(2,3),(2,4)}, {(1,4),(2,3),(2,4)}, {(1,3),(1,4),(2,3),(2,4)}.

AnswerA × B = {(1,3),(1,4),(2,3),(2,4)}; it has 16 subsets (listed above).

Watch this explained “Counting relations is counting subsets”, 6:55 into A relation is nothing more than a chosen part of the product

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1. Every second entry of the given pairs must be in B: the second entries are 1, 2, 1, so B contains 1 and 2.
2. n(B) = 2 and B already has 2 elements, so B = {1, 2}.
3. Every first entry of the given pairs must be in A: the first entries are x, y, z.
4. x, y, z are distinct, so that is already 3 different elements, matching n(A) = 3, so A = {x, y, z}.

AnswerA = {x, y, z}, B = {1, 2}

Watch this explained “Reading a product backwards”, 10:55 into Counting all the pairs, and why swapping the two sets gives something else

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1. If A has n elements, A × A has n² elements. Here n² = 9, so n = 3: A has 3 elements.
2. The two given pairs (−1, 0) and (0, 1) mention 3 different numbers: −1, 0 and 1.
3. Since A must have exactly 3 elements, A = {−1, 0, 1}.
4. List all 9 pairs of A × A and remove the 2 that were already given.

AnswerA = {−1, 0, 1}; the remaining 7 elements of A × A are (−1,−1), (−1,1), (0,−1), (0,0), (1,−1), (1,0), (1,1).

Watch this explained “From the count alone”, 12:03 into Counting all the pairs, and why swapping the two sets gives something else