# File

```1. Define Discrete Math. Give the reasons for studying it.
2. There are 2000 university students in total. 65 students take math. 80 students take physics. 1885
students take neither math nor physics. How many students take both math and physics? How many
students take only math? How many students take only physics? Draw the Venn’s diagram.
3. Find cardinalities and powers of each of these sets.
a. {6, 2, 1, 6, 0}
b. {3, 2, 5, 8, 9, 11, 5, 3, 4}
4. Give the order of the sets operations and logical operations.
5. Define a proposition, atomic proposition and compound proposition.
6. Link sets and propositional operations.
7. Give arithmetic equivalents of sets and propositional operations.
8. Prove.
a. A  A  B = A
b. A  (A  B) = A
c. (A  B  C)’ = (C’  B’)  A’
9. Define quantum logic, fuzzy logic and Boolean logic.
10. Give the truth tables for the main logical operations.
11. Give the truth tables for these.
А  (&not;B  C)
&not; (А  B)  (A  &not; B)
(А  B)  (C  B)
12. Is (&not;(&not;А)
 С)  B  (&not;C) equivalent to
1) A  (&not;C),
2) (&not;A)  B,
3) A  (&not;C)?
13. Explain tautology, contradiction, contingency, syllogisms, and predicates. Give examples.
14. Give and explain De Morgan’s laws. Why are they used?
15. Explain the main methods of mathematical proof.
16. Give the examples of direct, by contraposition, and by induction proofs.
17. Prove that √2 is an irrational number.
18. Define relations. Give examples of the matrices.
19. Find the pattern and write 2 more terms of each of these sequences.
a. 6, 12, 18, . . .
b. 1, 3, 6, 10, . . .
20. Represent each of these decimal numbers in binary and hexadecimal counting systems.
a. 67
b. 94
21. Give the best algorithm of finding the largest prime number.
22. Explain the public key cryptography.
23. Define combinations and permutations.
24. The TOEFL score is calculated as S = (S1+S2+S3)*10/3. How many TOEFL scores are possible?