CMSC 175 Discrete Mathematics
Final Exam Study Guide
The final exam is comprehensive. It will include
- writing definitions – you should know all definitions given in the lessons.
- multiple choice questions – review the multiple choice questions in the unit
tests.
- problems to solve – review all problems in the homework assignments and the
unit tests
The final exam will not include Boolean Algebra
A. Logic
You should know: truth tables, logical equivalences, de Morgan’s laws, conditionals
(contrapositive, converse, inverse, disjunctive representation, negation of a conditional,
representation of “unless” and “only if”, necessary and sufficient conditions), negation of
quantifies expressions, arguments in propositional and predicate logic. See all problems in the
homework assignments and Test 1
Sample problem: Show how t can be derived:
Premises:
1. p V q
2. q  r
3. p  s  t
4. ~r
5. ~q  u  s
6. Conclusion: t
B. Proofs
See all proof problems in the homework assignments and tests. Sample problems:
Choose and apply a method of proof or disproof (direct proof, proof by contraposition,
proof by contradiction, disproof by counterexample, proof by mathematical induction) to
prove or disprove the following statements:
1. For all integers n, n2 is even if and only if n is even. Note: if and only if means: a)
if n2 is even then n is even, b) if n is even then n2 is even
2. For all integers k, k3 is odd if and only if k is odd
3. 2 + 4 + 6 + …. + 2n = n(n+1) , for all n ≥1
4. 12 + 32 + 52 + …. + (2n-1) 2 = n(2n+1)(2n-1)/3, n ≥1
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5. 1 + a + a2 + a3 + …. + a(n-1) = (an – 1)/(a-1), n ≥1, a  1
6. For all integers n, if n is even then (n-1)(n+1) is odd
7. For all integers n, if (n+1)(n-1) is odd then n is even
8. Consider the sequence a1, a2, …, an …. defined recursively:
a1 = a
an+1 = an + d
Prove that an = a1 + (n-1)d
9. The square of any integer can be written in one of the following forms: 4k or
4k+1
10. Let n be an odd integer. Then n3 + 2n2 is also odd.
11. 2 + 6 + 18 + …. + 2*3(n-1) = 3n – 1, n ≥1
12. 1*2 + 2*3 + 3*4 + …. + n(n+1) = n(n+1)(n+2)/3
13. 32n + 7 is divisible by 8 for all n ≥1
14. 11n – 6 is divisible by 5 for all n ≥1
15. 6*7n – 2*3n is divisible by 4 for all n ≥1
Prove
(A  B) – B = A – B
(A – B)  B = A  B
A – (A – B ) = A  B
A – B = A  ~B
A  (~A  B) = A  B
(A – B)  (B – A) = (A  B) – (A  B)
A – (A  B) = A ~B
(A  B )  (A  ~B ) = A
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C. Relations
See all problems on relations in the homework assignments and in the tests.
Sample problems:
1. Let X = {1,2,3,4,…, 10}. Define a relation R on X x X :
R = { ((a, b), (c, d)) | a + d = b + c}
Show that R is an equivalence relation on X x X
2. Let X = {1,2,3,4,…, 10}. Define a relation R on X x X :
R = { ((a, b), (c, d)) | ad = bc}
Show that R is an equivalence relation on X x X
3. Let R and S be two relations on a set A. Given the properties of R and S, determine
the corresponding properties of R S, R  S, ~R.
D. Recurrence Relations
Solve the recurrence relations below with initial condition T(1) = 1:
1. T(N) = T(N-1) + 1
2. T(N) = T(N-1) + N
3. T(N) = 2T(N-1) + 1
4. T(N) = T(N/2) + 1
5. T(N) = T(N/2) + N
6. T(N) = 2T(N/2) + 1
7. T(N) = 2T(N/2) + N
E. Counting
See the problems in Lesson 21: Counting. Some more problems:
1. A select committee of three persons is to be formed from five Conservative, three
Labor and four Liberal Democrat members.
a. In how many ways can the committee be formed?
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b. In how many ways can the committee be formed if at least one Liberal
Democrat must be included?
c. In how many ways can the committee be formed if it cannot include both
Labor and Conservative members?
d. In how many ways can the committee be formed if it must include at least
one conservative and one Labor member?
2. A palindrome is a string that reads the same backwards as forwards. Consider
palindromes of digits only (decimal digits 0, 1, 2, …, 9). How many different
palindromes are there of length 6? Of length 7?
3. How many 4-digit numbers less than 6000 (decimal number system) can be built
using only odd digits? Assume that numbers do not start with leading zeros.
4. How many 3-digit hexadecimal numbers are there less than F00, but greater than
999? How many of them end in 1?
5. A computer password consists of 8 characters. Allowed characters are: Letters in
the English alphabet - lower and upper case, decimal digits 0, 1, .., 9, and the
underscore symbol. The first two characters must be lower case letters.
a. How many different passwords are possible?
b. How many passwords with no repeated characters are possible?
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