Difference and complement: the same idea with and without a universal set

What to assume they know

• Union and intersection, and what it means for two sets to miss each other entirely — union and intersection, and their properties
• Why nothing can be complemented until the surrounding set is fixed — the universal set as a choice made by context
• A set with nothing in it, and sets you cannot finish listing — the empty set
• Turning a claim about sets into a picture you can read off — reading a shaded Venn diagram
• Prime numbers; divisors of a two-digit number; rational and irrational numbers

What they should be able to do

• Form the difference of two sets in either order and show that the two results differ
• State the difference in set-builder form and identify the two conditions in it
• Explain why difference is not commutative while union and intersection are
• Decompose a pair of sets into three non-overlapping pieces and say why they cannot overlap
• Define the complement relative to a stated universe, and compute it
• Express the complement as a difference and use that to explain each complement law
• Compute complements over the natural numbers for conditions given in words
• Name which complement laws have no counterpart for difference, and say why

Where it usually goes wrong

• "A minus B is the same as B minus A." Example 18 gives three members one way and one the other.
• "Order does not matter for set operations." It does not for union and intersection, because their conditions treat the two sets alike. Difference's condition does not — one set is entered, the other is avoided.
• "A minus B removes the elements of B." It removes only those of B that were in A. The 8 in Example 18 was never in A and nothing happens to it.
• "The complement of a set is everything else in the world." Everything else in the chosen universe. Example 21 is built to show the answer moving when the universe does.
• "An endless set must have an endless complement." The chapter's own first complement example has an endless set whose complement has three members.
• "Difference should have the same laws as complement." It cannot. There is no fixed left operand, so nothing plays the role the universe plays.
• "The complement of the complement is something new." It is the original set, and the chapter prints that as a law of its own.
• "Non-primes over the naturals means the composite numbers." It means the composites together with 1, since 1 is a natural number and is not prime.

Questions to check understanding

• Form both differences of a given pair and comment on the comparison
• Find the complement of a set under a stated universe
• Complement a condition given in words over the natural numbers
• Fill in the blanks in the complement laws
• Verify on given sets that the three pieces of a pair do not overlap and that together they give the union
• Say what a stated complement becomes if the universe is changed

Examples worth working on the board

Values marked verified are an added derivation from the chapter's own data; the chapter prints no answers to its exercises.

• Example 18 (p. 16). A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. Verified: removing B from A leaves {1, 3, 5}; removing A from B leaves {8}. Three members against one — the asymmetry is visible at a glance, which is why this example comes first.
• Example 19 (p. 17). V is the five vowels and B = {a, i, k, u}. Verified: removing B from V leaves {e, o}; removing V from B leaves {k}. A second pair making the same point with letters.
• The set-builder form (§1.9.3, p. 17). The difference is written with two conditions joined: membership of the first set, and non-membership of the second. Section 3's argument lives here — swap the two sets and you swap which condition is the positive one.
• The three-piece decomposition (Remark, p. 17). The chapter states that A-minus-B, the intersection, and B-minus-A have no member in common with each other. Verified: an object in A-minus-B fails the test for B, so it cannot be in either of the other two, and the same argument runs for each pair. Together the three pieces make up the union, which is Miscellaneous Exercise question 6 on p. 21.
• Fig 1.8 (p. 17) and Fig 1.9 (p. 17). Read off the printed page: Fig 1.8 shows two overlapping circles inside a rectangle labelled U, with only the part of the left circle outside the right one tinted and labelled at the lower left. Fig 1.9 repeats that drawing unchanged — same two circles, same positions, same overlap, measured on a single close-up holding both at one scale — and adds tints and labels: the left crescent in a light tint labelled with an arrow at the lower left, the lens in a darker tint labelled beneath with an arrow, and the right crescent untinted with its own label and arrow coming in from the upper right. Fig 1.9 is the picture of section 4 and the three arrows are the three pieces.
• The chapter's opening complement case (§1.10, p. 18). The universe is taken to be all the prime numbers and A is the primes that do not divide 42. Verified: 42 = 2 × 3 × 7, so the primes dividing it are 2, 3 and 7, and those three are exactly what lies outside A. Note the shape of this: A is endless and what lies outside it has three members. That only makes sense once the universe is in view, which is section 7's point.
• Example 20 (p. 19). The universe is 1 to 10 and A holds the odd ones. Verified: outside A are 2, 4, 6, 8 and 10.
• Example 21 (p. 19). The universe is one coeducational Class XI and A is the girls in it; outside A are the boys. Change the universe and the answer changes, which is why this example is worth keeping.
• Fig 1.10 (p. 19). Read off the printed page: a rectangle tinted throughout except for a white circle labelled A, with the complement's label set in the rectangle's upper right corner.
• The four printed complement laws (§1.10, p. 20). A set together with its complement makes the universe; a set met with its complement is empty; complementing twice returns the set; and the empty set and the universe are each other's complements. The chapter also prints De Morgan's pair in the same list, which is taught in Why complementing turns each of the two operations into the other.
• Exercise 1.4 Q9 to Q11 (p. 18) — the difference exercises. (Q12 on the same page is four true-or-false claims about disjointness and belongs to Union and intersection, and what it means for two sets to miss each other entirely.) A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, with twelve differences to form; then X = {a, b, c, d} against Y = {f, b, d, g}; and then the reals with the rationals removed. Verified: removing B from A leaves {3, 6, 9, 15, 18, 21} and removing A from B leaves {4, 8, 16, 20}; removing Y from X leaves {a, c} while removing X from Y leaves {f, g}, with the shared part {b, d}; and the reals minus the rationals are the irrationals, which is the chapter's own letter T from §1.6.1.
• Exercise 1.5 Q1 and Q2 (p. 20). With the universe 1 to 9, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}: verified: outside A are 5, 6, 7, 8, 9; outside B are 1, 3, 5, 7, 9; outside the union of A and C are 7, 8, 9; outside the union of A and B are 5, 7, 9; complementing A twice returns A; and removing C from B leaves {2, 8}, so outside that are 1, 3, 4, 5, 6, 7 and 9. Question 2 repeats the drill over an eight-letter universe.
• Exercise 1.5 Q3 (p. 20) — eleven conditions, complemented over the naturals. Verified: the even and the odd naturals are each other's complements; outside the positive multiples of 3 are the naturals that 3 does not divide; outside the primes are 1 together with the composite numbers — 1 is the item students drop; outside the naturals divisible by both 3 and 5 are those not divisible by 15; outside the perfect squares are the non-squares, and likewise for cubes; the condition x + 5 = 8 gives {3}, so its complement is every natural except 3; 2x + 5 = 9 gives {2}, so its complement is every natural except 2; the naturals at least 7 have complement {1, 2, 3, 4, 5, 6}; and 2x + 1 > 10 forces x above 4.5, hence x at least 5, so that complement is {1, 2, 3, 4}.
• Exercise 1.5 Q6 and Q7 (p. 20). With the universe all the triangles in a plane and A the triangles having at least one angle other than 60°, verified: outside A are the triangles whose three angles are all 60°, that is the equilateral ones. Question 7 asks for four fill-ins, which are the complement laws restated: the union with the complement, the universe met with a set, the intersection with the complement, and the empty set met with a set.

Figures to have open

• Redraws of Fig 1.8 and Fig 1.9 (p. 17). Fig 1.9 carries section 4 and must keep its three labelled regions; redraw as clean schematics rather than reproducing the printed art.
• A redraw of Fig 1.10 (p. 19) for sections 5 and 7, with the rectangle tinted and the circle left clear.
• A paired-rectangle diagram for section 7: the same circle drawn inside two different universes with the leftover regions labelled differently. An added standard schematic.
• A number-line or list device for Exercise 1.5 Q3, showing the naturals with the described set struck out and the survivors highlighted. Standard schematic.

Where this sits in the book

• NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.9.3 Difference of sets, pp. 16–17, including the Remark on p. 17, and §1.10 Complement of a Set, pp. 18–20, including Definition 7
• Worked Examples 18 and 19, pp. 16–17; Examples 20 and 21, p. 19
• Fig 1.8 and Fig 1.9, p. 17; Fig 1.10, p. 19
• The list of complement properties, p. 20
• Exercise 1.4, questions 9 to 11, p. 18; Exercise 1.5, questions 1, 2, 3, 6 and 7, p. 20
• Miscellaneous Exercise questions 5 and 6, p. 21, which are the proof versions of sections 3 and 4