Relations and Functions: questions with answers

Line 2 is where the rule is applied, before any arithmetic is done: corresponding entries are equated, first with first and second with second. Pairing x + 1 with 5 crosses the components; writing only the first equation stops halfway and leaves y undetermined; adding the entries together abandons the pair structure altogether.

(6, 3) takes its first entry from A and its second from B and is not among the three already written. (2, 6) puts an element of B first, so it belongs to B × A and not to A × B at all, and (6, 2) is already listed.

The first entry 3 can pair with each of B’s three elements, giving three pairs. Pairing only matching list positions would give just one pair with this first entry.

If n(A) = p and n(B) = q then n(A × B) = pq, so the answer is 4 × 2 = 8 pairs. Adding the sizes counts a single choice, but here a first element and a second element are chosen independently, which multiplies. Reporting the larger set's size forgets that each of its elements is used many times over, and doubling counts each pair in both orders when only one order belongs to A × B.

One entry is needed for each factor, so 2 numbers name an element of R × R. Answering with a single number treats the product as though it were R itself, which is exactly the step that turns the plane back into a line.

The law reaches both terms: A × (B ∩ C) = (A × B) ∩ (A × C). Leaving the second term alone gives (A × B) ∩ C, which is not even the same kind of object — one side of the connective is a set of PAIRS and the other a set of loose elements. Applying the connective between A and each of B and C changes what is being multiplied, and multiplying the two products together builds pairs of pairs.

Each factor has 5 elements, and every first choice pairs with every second choice. Thus the product contains 25 pairs. Counting only 5 uses one factor’s size. Counting 5 × (5 − 1) leaves out pairs with matching entries.

Undoing the rule gives x = −2: the constant comes off first, then the multiplier. Reporting 2 drops the sign, and f(2) would be 5, not -15. The two remaining answers skip the division by 5 altogether.

Only the x with 4x ≤ 15 give a pair, and there are 3 of them. Answering 15 assumes every element of A must appear as a first entry, which the definition never demands, and 15 × 15 counts the whole of A × A, which is what the rule was there to narrow.

Only the three elements that start an arrow belong to the domain, so the domain is {10, 11, 12} and 13 is left out even though it is drawn in the oval. The set the arrows LAND on is the range, not the domain, and the whole left oval is the set R was drawn from rather than the part of it R actually uses.

The failure is that 3 carries two images. It is also true that 5 is shared by 3 and 4, but a shared IMAGE is not a fault at all — a function may be many-to-one, and the chapter's own answer confirms {(2,1), (5,1), ...} is a function with range {1}. Picking the shared image names something true and blames the wrong thing.

Counting relations is counting SUBSETS of A × B. There are 2 × 3 = 6 pairs, and each is independently in or out, so the total is 2^6 = 64 relations. Stopping at 6 counts the pairs themselves, and doubling them applies the choice once instead of once per pair. The exponent is the number of PAIRS, so it is 2 × 3 and not 2 + 3.

Division fails only when the denominator is zero, so exactly 4 and 8 are removed. The numerator's zeros 10 and 13 are perfectly good inputs — there f simply takes the value 0, which is a value like any other. Removing all four points throws away two inputs the rule handles happily, and removing none leaves the rule undefined at two of its own.

Every term of g is subtracted, so its constant changes sign along with its x term and the answer is 3x + 5. Leaving g's constant as it stands gives 3x - 1. Taking the difference the other way round gives -3x - 5 — the negative of the right answer throughout, since f − g and g − f are not the same function.

The drawn line tells you everything: level means the output never changes, which is a constant function whose range is one single value, and a 45-degree line through the origin means the output equals the input, which is the identity function. Calling a constant function's range the whole real line confuses the domain with the range. And the signum function's graph is not a single line at all — it is in three separate pieces, at the heights −1, 0 and 1.