Exam 2 - CMSC203 - Discrete Structures - Spring 2001
1. Circle T of the corresponding statement is True and F if it is False:
T F 1 + 2 + 3 + 4 + ... + 1,000 = 1,010
T F If A is a non-empty set, then A x A is an Equivalence Relation.
T F If A is a non-empty set, then A x A is function.
T F If f:X → Y is a function and f(X) = Y, then f is a ONE-TO-ONE function.
T F If A is a non-empty set, then ∅ is the smallest Equivalence Relation on A.
T F |N| = |Q|.
T F If f and g are functions with f:X → Y and g:Y → Z, then f ° g = g ° f.
T F If f:A → B is a function and iB:B → B is the identity function on B, then (iB ° f) = f.
T F The Weak Form of Mathematical Induction and the Strong Form are equivalent.
T F If R is an Equivalence Relation on a set A, and a,b ∈ A with [a] = [b], then (a,b) ∈ R.
2. Circle F for function, I for one-to-one, S for onto, and B for one-to-one correspondence as the properties apply
to the relations below. More than one choice or no choices may apply to a given relation.
3. Write in summation of i notation (starting with i = 1), then rewrite as a summation
ranging from j = 9 to 26.
4. Let Σ = {0,1}and let H be the Hamming distance function on binary strings. Consider the relation:
R = {(s,t) | s,t ∈ Σ4 and H(s,0000) = H(t,0000)}.
a. Prove that R is an Equivalence Relation. b. What partition of Σ4 does the relation R induce?
5. Let f:R → R be the function f(x) = 9x + 5. a. Show that that f is 1-1 and onto. b. Find f -1(x).
6. Let f = {(1,9),(2,7),(3,5),(4,3),(5,1)} and let g = {(1,8),(3,6),(5,4),(7,2),(9,0)}.
Show ( g ° f ) −1 = ( f −1 ° g−1 ).
7. Prove 1 of the following 2 statements using the indicated method:
a. Using Strong Induction, show that if n is an integer greater than 1, then n has a prime factor.
b. Using Weak Induction, show that 1 + 2 + 22 + 23 +...+ 2n = 2n+1 − 1.
a2
13
-----a3
24
-----a4
35
-----…a19
1820
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a. F I S B b. F I S B c. F I S B d. F I S B e. F I S B
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