Assignment 2 comp2130 Winter 2014. Due Feb. 14, 2014 Obey all instructions given in the roass document

Assignment 2 COMP2130 Winter 2014. Due Feb. 14, 2014
Obey all instructions given in the ROASS document.
Q1) Find the truth set for each of the following:

a) predicate: 7/d is an integer, d  Z⁺.

b) predicate: is an integer, d  { -1, 0 ,1, 5, 25}.

c) predicate : 1 < x²  3, x R.

Q2) Let D be the set of all students at your school, and let M(s) be “s is a math major” and C(s) be “s is a computer science major” and E(s) be “ s is an engineering student”. Express each of the following statement using quantifiers, variables and M(s), C(s), and E(s).

a) No engineering students are also math students or computer science students.

b) Some math students are computer science students but no engineers are computer science students.
Q3) Write in logical form: “the sum of any two fractions is an integer.”
Q4) Which of the following are a negation for: “All dogs are loyal”. Be careful there may be more than one correct answer.

a) All dogs are disloyal.

b) No dogs are disloyal.

c) Some dogs are loyal.

d) Some dogs are disloyal.

e) There exists disloyal animals that are not dogs.

f) There exists loyal animals that are dogs.

g) No animals that are not dogs are loyal.

h) Not all dogs are loyal.

i) There is an animal that is not a dog but is loyal.

Q5) Is the following statement true? “For all real numbers, x and y, if x²=y² then x = y.” What is its negation? Is the negation true?
Q6) Consider the sequence: 1415926535. “All the 3’s immediately following a 1 are followed by a 7” Is this statement true? Why?
Q7) Rewrite the following in the if-then form: “Being on time each day condition is a sufficient condition for passing this course.”
Q8) Let D=E={-2.-1,0,1,2}. Explain why the following are true or false.

a) x  D, y E such that x²-y²  2.

b) xD such that y  E, x+y= -1.
Q9) Write the negation for Q8a) and Q8b.
Q10) Fill in the blank in these argments.

a) Students in this class are intelligent.

Joan is not in this class.

___________________

b) Students in this class are intelligent.

Joan is not intelligent.

____________________________
c) Students in this class are intelligent

Joan is in this class.

__________________________________
d) Students in this class are intelligent

Joan is intelligent.

__________________________________

Q11) Put each statement into logic form and then rearrange the new statements into the proper order. You may need to use the contrapositive form. Give the conclusion.

1) No one, who is going to a party, ever fails to brush his teeth;

2) No one looks cool, if he is untidy;

3) Marijuana smokers have no self-control;

4) Every one, who has brushed his teeth, looks cool;

5) No one wears a costume, unless he is going to a party;

6) A man is always untidy, if he has no self-control;

Q12) Prove the following theorem: For a,b,c,m Z, if m|a and if m|b and a+b+c = 0, then m|c.

Q13) Prove that the product of an even integer and an odd integer is even.

Q14) What is the largest positive integer that the product of 5 consecutive integers is always divisible by? Don’t prove it unless you want to. There are no marks for it.

Q15) Prove or disprove that if a non-zero rational number is divided by an irrational number, then the result is irrational.