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Sample Exam 2 - Fall 1999 - CMSC 203 / Discrete Structures
Symbols: N denotes the Natural Numbers, Z denotes the Integers, Q denotes the Rational Num-
bers, and R denotes the Real Numbers.
1. Circle T if the statement is true or F if the statement is false.
T F If A is a non-empty set, then ∅ is a the smallest equivalence relation on A.
T F If A is a non-empty set, then A×A is the largest equivalence relation on A.
TFLet R = (A×A) ∪ (B×B) ∪ (C×C) ∪ (D×D) is an equivalence relation on a set X.
Then the sets, A, B, C, and D, partition X .
T F One-to-one functions map larger finite sets into smaller finite sets.
T F If a function is onto, its range (co-domain) equals its image.
TFIf f:A → B and g:B → A are 1-1 and onto functions, then g ° f = f ° g .
TFIf n is a positive integer, 9(1 + 10 + 102 + ... + 10(n−1) ) = (10n − 9).
T F The Weak and Strong Forms of Mathematical Induction are equivalent.
T F If H is the Hamming distance function, then
H( 111000 , 000000 ) = H( 111000 , 111111 ).
T F There are as many prime numbers as rational numbers.
2. Rewrite: as a sum from 6 to 15.
3. Let R = {(a,b) | a,b ∈ {1,2,3,4,5} and a + b < 5}. Graph R.
2 •
1 • • 4
3 • • 5
4. Prove 1 of the 2 Theorems below by Induction:
Theorem: A set with n elements has 2n subsets.
Theorem: Every integer greater than 1 can be factored as the product of prime numbers.
5. Let f = {(1,3),(2,1),(3,5),(4,4),(5,2)} and g = {(1,2),(2,3),(3,4),(4,5),(5,1)}.
Show that ( g ° f ) −1 = f −1 ° g −1.
1
12⋅
---------- 2
23⋅
---------- 4
34⋅
---------- 8
45⋅
----------16
56⋅
---------- …512
10 11⋅
----------------++++++
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