Relations and Functions — Questions & Answers

Sample questions from Relations and Functions

1. Q1. If f: R → R is continuous and f(f(x)) = x for all x, and f is not the identity, then:

   • A.f must be linear
   • B.f must decrease on some interval✓
   • C.f must be strictly increasing everywhere
   • D.f must be a constant
   Solution

   B is correct. A continuous involution f: R→R with f≠I must be strictly decreasing (a result from real analysis). A non-identity continuous involution on R swaps points in a decreasing pattern: if f were increasing, then f(f(x))>x for f(x)>x, contradicting f(f(x))=x. So f must be strictly decreasing (hence decreasing on the whole real line).

2. Q2. The number of onto functions from a set of 2 elements to a set of 2 elements is:

   • A.1
   • B.2✓
   • C.4
   • D.0
   Solution

   B is correct. Onto functions from {a,b} to {1,2}: (a→1,b→2) and (a→2,b→1). That is 2 functions. In general, surjections from n-element set to n-element set = n! = 2! = 2.

3. Q3. Let A = {1,2,3,4,5,6}. The number of ordered pairs of functions (f, g) from A to A with g o f = identity on A is:

   • A.1
   • B.6^6
   • C.36
   • D.720✓
   Solution

   g o f = identity on a finite set of the same size forces f to be a bijection, and then g must be f inverse, uniquely determined. So the number of pairs equals the number of bijections, 6! = 720.

4. Q4. Let f: Z x Z -> Z be f(m,n) = 3m + 5n. Then f is:

   • A.one-one but not onto
   • B.onto but not one-one✓
   • C.a bijection
   • D.neither one-one nor onto
   Solution

   For any integer k, f(2k, -k) = 6k - 5k = k, so f is onto. It is not one-one since f(5, -3) = 15 - 15 = 0 = f(0, 0) while (5,-3) differs from (0,0).

5. Q5. Let A = {1, 2, 3, 4, 5, 6, 7}. The number of equivalence relations on A whose classes all have size 1 or 2, with precisely two classes of size 2, is:

   • A.42
   • B.21
   • C.210
   • D.105✓
   Solution

   Choose the 4 elements that get paired: C(7,4) = 35 ways. Split those 4 into two unordered pairs: 3 ways. The remaining 3 elements are forced to be singleton classes. Total 35 x 3 = 105.

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