Revision paper in discrete math for the Mid-Term Exam:
1.1. Give cardinalities and powers of each of these sets. a. {6, 2, 1, 6, 0} b. {3, 2, 5, 8, 9, 11, 5, 3, 4}
1.2. Find Cartesian product of these sets {p, g, a} and {7, 3, 1, 6}.
2.1. Give order of logical operations, truth tables and arithmetic equivalents for main logical gates.
2.2. Explain proposition, tautology, contradiction, contingency, syllogisms, and predicates. Give examples.
3.1. Prove that √2 is irrational number and Triangular Number expression.
3.2. Give converse, inverse and contrapositive to “If I study hard, then I am rich.”
4. Find f(f(f(f(f(2))))) if f(x) = x2.
5.1. Binary relation R on the set {1, 2, 3, 4} is defined so that aRb holds if and only if
a divides b, with remainder. Find the matrix and draw the graph.
5.2. Do men or women marry more? Explain using Discrete Math.
5.3. Explain recursion and iteration. Why are prime number and factorization important?
6.1. Represent each of these decimal numbers in numeral systems with bases 2,5,7,9,16. a.67
b.94
6.2. Calculate the largest prime number. Give prime factorization of your student number.
7.1. Find the number of grains for the Chess problem.
7.2. How many up to 6-symbols passwords can be made of 26 letters (a-z) and 10 digits (0-9)?
7.3. In how many ways can we answer TOEFL? How many lines are needed to connect P points?
7.4. How many options are there for a password made of 5 digits, selected from 6 digits?
8.1. Prove these properties of combinations and permutations.
a.C(n,k)=C(n,n-k)
b.P(n,k)=C(n, k)P(k, k)
c.C(n,k)+C(n,k+1)=C(n+1,k+1) d.P(n,n)=P(n,n-1)=n!
8.2. Calculate
a. ∑𝑛𝑘=0 𝐶(𝑛, 𝑘)
b. ∑𝑛𝑘=0(−1)𝑘 𝐶(𝑛, 𝑘)
9. Solve Sokoban, Scrabble, Factorial, Prime, Number, Password and Graceful Graph puzzles.
10. Describe your project.
Deadline: 5.5.2015