Final Exam Study Guide Ma 320-CRN 22189 Spring 2106 Carter
Summary I expect to be asking questions in the penumbra of the questions listed here.
1. p. 5 #1.42,
2. p. 5 #1.45,
3. p. 9 #2.24
4. p. 9 #2.25,
5. p. 12 #3.14,
6. p. 13 #3.18,
7. p. 14 #3.29,
8. p. 15 # 3.36
9. p. 15 # 3.43
10. Consider a set X = {1, 2, . . . , n} and a set Y = {1, 2, . . . , m}.
(a) What is the number of elements in the Cartesean product
X × Y = {(x, y) : x ∈ X & y ∈ Y }?
(b) What is meant by a relation from X to Y ?
(c) How many relations are there?
(d) Prove this.
f
(e) What is meant by a function Y ←− X?
(f) How many functions are there from X to Y ?
(g) Prove this.
(h) If m < n, are there any injective function?
(i) In case n < m how many injective functions are there?
(j) Prove this.
(k) If n < m are there any surjective functions.
(l) Be able to define injective and surjective (one-to-one and onto respectively).
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(m) In case m = 2, let’s say Y = {0, 1}. What is the correspondence between
f
{{0, 1} ←− {1, 2, . . . , n}} and the set of subsets of {1, 2, . . . , n}? Describe this
explicitly, and explain why the power set has 2n elements.
(n) In case, n = m, prove that there are n! bijections from X to Y .
11. Extra credit: prove the binomial theorem:
n X
n n−k k
(x + y) =
x y .
k
k=0
n
12. What is the most interesting thing that you learned this semester?
13. What is one thing that you wish you understood more?
14. Why do you think we should master the ability to prove or disprove mathematical
statements?
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