What goes in, what comes out, and what was merely allowed to come out

What to assume they know

• A relation is nothing more than a chosen part of the product — a relation as a chosen subset of the product
• Why writing a pair in order carries information a set cannot — first and second entries of an ordered pair
• Subset and containment notation, and the difference between ⊂ and =
• The natural numbers, the integers, and divisibility of one whole number by another
• Substituting a value into a simple linear condition and checking whether the result belongs to a stated set

What they should be able to do

• Read a relation's domain, and its range, directly off a roster of pairs
• Read both of those sets off an arrow diagram, including cases where an element of either oval is unattached
• State where the codomain comes from, and explain why it cannot be computed from the pairs
• Explain why Definition 4 asserts containment and not equality, and give an instance from the chapter where the containment is strict
• Work out all three sets for a relation given as a condition, testing candidate values for membership in the stated sets
• Describe what changes and what does not when the second set is replaced by a smaller one that still holds every image
• Explain why a relation is free to leave elements of the first set unused, and flag that this freedom is what §2.4 will take away

Where it usually goes wrong

• "Range and codomain are two words for the same set." Definition 4 writes containment. Fig 2.6 shows an element sitting in the codomain that the range does not contain.
• "The codomain is whatever set the images happen to fill." Then it would be the range. It is the set you named when you set the relation up, and Exercise 2.2 Q1 declares a fourteen-element codomain for a relation whose range has four elements.
• "The domain must be the whole first set." For relations it need not be — Example 7 leaves 6 out. This is precisely the freedom that Definition 5 removes when it defines a function, so the misconception is worth planting here and correcting there.
• "If a condition is stated on a set, every element of that set is in the domain." Exercise 2.2 Q1 refutes it: the condition sends 5 to 15, which is not in the set, so 5 never opens a pair.
• "An unattached element means the diagram is wrong." It means the range is strictly inside the codomain, which is the ordinary situation.
• "Shrinking the second set changes the relation." The pairs are unchanged; what changes is what the codomain records. The chapter does not discuss this, so present it as a question the definitions answer rather than as a printed result.
• "Domain and range are always the same size." Exercise 2.2 Q5 has both equal to a five-element set while the relation has twelve pairs, and Example 8 has a three-element domain against a six-element range.

Questions to check understanding

• Given a relation in roster form, state its domain and its range
• Given a relation as a condition on a stated finite set, work out which elements actually open a pair, and give domain, codomain and range
• Given an arrow diagram, give all three sets
• Explain why the range of a stated relation is a proper part of its codomain
• Decide whether a stated element belongs to the domain, the range, both or neither
• Name both computed sets for a relation on an infinite set where the condition turns out to hold everywhere

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data.

• Example 7 (pp. 28–29). A holds 1 through 6, and the relation from A to A keeps the pairs whose closing entry is one more than the opening entry. Verified: the five pairs are 1 with 2, 2 with 3, 3 with 4, 4 with 5, 5 with 6. The domain is 1 to 5, the range is 2 to 6, and the codomain is all six. So 6 belongs to the codomain and the range but not the domain, and 1 belongs to the domain and the codomain but not the range. That single relation exhibits both kinds of shortfall at once, which makes it the best example in the chapter for this topic.
• Fig 2.5 (p. 29). Read from the printed page: two ovals each carrying 1 through 6, five arrows running between them, and neither 6 on the left nor 1 on the right attached to anything.
• Example 8 and Fig 2.6 (p. 29). Read from the printed page: the left oval holds 9, 4 and 25; the right oval holds 5, 3, 2, 1, −2, −3 and −5; six arrows. The chapter's reading is that the left entry is the square of the right one. Verified: domain is 4, 9 and 25 — the whole left oval; range is 2, −2, 3, −3, 5 and −5 — six of the seven right-hand elements. The codomain is the whole right oval, and 1 sits in it unattached, because 1 squared is 1 and 1 is not in the left oval. This is where containment is strict and visibly so.
• Exercise 2.2 Q1 (p. 29). A holds 1 through 14, and the relation from A to A keeps the pairs for which three times the opening entry equals the closing entry. Verified: the openings that work are 1, 2, 3 and 4, giving closings 3, 6, 9 and 12; the opening 5 would need 15, which is outside A, and every larger opening fails likewise. Domain is 1 to 4, range is 3, 6, 9, 12, codomain is all fourteen.
• Exercise 2.2 Q2 (p. 30). On the natural numbers, the closing entry exceeds the opening by 5, and the opening is a natural number below 4. Verified: the openings are 1, 2 and 3, giving 6, 7 and 8. Domain 1, 2, 3; range 6, 7, 8.
• Exercise 2.2 Q5 (p. 30). A holds 1, 2, 3, 4 and 6, and the relation keeps a pair when the closing entry is exactly divisible by the opening one. Verified by testing all 25 pairs: 1 divides all five, so five pairs; 2 divides 2, 4 and 6, three pairs; 3 divides 3 and 6, two pairs; 4 divides only 4, one pair; 6 divides only 6, one pair. Twelve pairs in all. Domain is all of A and range is all of A, so here the containment is not strict — a useful contrast with Example 8.
• Exercise 2.2 Q6 (p. 30). The openings are 0 through 5 and each closing entry is 5 more than its opening. Verified: domain 0 to 5, range 5 to 10.
• Exercise 2.2 Q9 (p. 30). On the integers, a pair is kept when the difference of its two entries is an integer. Verified: that is true of every pair of integers, so the relation is the whole product, and both domain and range are the integers. The point is that a condition can be vacuous, and you only find out by testing it.
• Fig 2.7, Exercise 2.2 Q4 (p. 30). Read from the printed page: left oval 5, 6, 7; right oval 3, 4, 5; three arrows dropping each entry by 2. Verified: domain is the whole left oval, range is the whole right oval, and the codomain is that same right oval — a case where all three could be confused because two of them coincide.

Figures to have open

• Fig 2.5 redrawn as a schematic (p. 29), with the unused element on each side distinguishable. Sections 4 and 6 depend on it.
• Fig 2.6 redrawn (p. 29), all seven right-hand elements present. The element list was read off the printed page and is not in the extracted text.
• Fig 2.7 redrawn (p. 30) for section 11's contrast case, again read off the page.
• A three-box panel that can hold domain, range and codomain and be refilled for each worked relation. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.3 "Relations", pp. 28–30.
• Definitions 3 and 4, p. 28.
• Examples 7 and 8, pp. 28–29.
• Exercise 2.2 items 1, 2, 4, 5, 6 and 9, pp. 29–30.
• Summary, p. 41, for the domain and range entries.
• Figs 2.5 and 2.6 (p. 29), Fig 2.7 (p. 30).