CPCS 222 – DISCRETE STRUCTURES
Mathematical Reasoning and Proof Exercises 3a-c
1. Translate these statements into English, where the domain of each variable consists of all real numbers.
a. ∃x ∀y (x < y)
b. ∀x ∀y ((( x ≥ 0) ∧ (y < 0)) → (xy ≥ 0)) d. ∀x ∀y ∃z (xy = z)
2. Interpret the result of Question 1(a) if p and q denote
a. logical expressions x and y b. sets A and B
3. Let, P(x,y) be a statement that “the student has taken class y” , where the domain consists of all the students in your class and for y consist of all computer science courses of your college.
Express each of these quantification in English.
a. ∃x ∃y P(x,y) b. ∃y ∀x P(x,y) c. ∀y ∃x P(x,y)
4. Let S(x) be the statement e “x is a student”, F(x) is the predicate “x is a faculty member” and A(x, y) is the statement “x has asked y a question”, where the domain consist of people in your college. Use quantifiers to express each of these statements
a. Usama has asked Nashwan a question.
b. Every student has asked Dr Samir a question.
c. There is someone in your class has either sent an e-mail message or telephoned everyone else in your class.
d. There are two different students in your class who have sent each other e-mail messages.
e. Every student in the class has either received an e-mail message or receives a telephone call from another student in your class.
f. There is a student has sent himself an e-mail message.
5. Find the argument form and for the following argument and determine whether it is valid. Can we conclude the conclusion is true when the premises are true?
If Dr Ali does not have eight legs, then he is not an insect.
Dr Ali is an insect.
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∴ Dr Ali has eight legs
6. What rules of inference is used in each of these arguments?
a. Abdullah is IT major. Therefore, Abdullah is either an IT major or CS major.
b. Muhammad is an IT major and CS major. Therefore, Muhammad is an IT major.
c. If it is rainy, we don’t go to barbeque. It is rainy; therefore we don’t go to barbeque.
d. If Mohammed Rafeeque is sick today there will be no Discrete Structure lab. There is a Discrete Structure lab today. Therefore, Mohammed Rafeeque is not sick.
e. If I come to all classes of CPCS 222, then I can answer all exercises. If I answer all exercises, then I understand the material. Therefore, If I come to all classes of CPCS 222,
then I will understand all the material.
f. Abdulaziz is a an excellent student. If Abdulaziz is an excellent student, then he
can earn his degree, has a secured job and get married. Therefore, Abdulaziz can earn his degree, has a secured job and get married.
7. Use direct proof to show that the sum of two odd integers is even.
8. Use direct proof to show that the sum of two even integers is even.
9. Give a direct proof of “ If n is odd integer, then n2 is odd”
10. Prof that if n is an integer and 3n + 2 is odd then n is odd.
11. Use proof by contradiction to prove that the sum of an irrational number and rational number is irrational
