Assignment No 1 (Solved)

Q1.    Let p, q, r be the propositions:             p: You have flu.             q: You miss the final examination.             r: You pass the course.             Express each of the following propositions as an English sentence.

1) p → q                                   3) q → ï¿¢r                               5) (p → ï¿¢r) ∨ (q → ï¿¢r)

2) ï¿¢q ↔ r                                4) p ∨ q ∨ r                               6) (p ∧ q) ∨ (ï¿¢q ∧ r)

Solution:

1. p → q

If you have flu, you miss the final examination.

1. ï¿¢q ↔ r

You pass the course if and only if you do not miss the final examination.

1. q → ï¿¢r

If you miss the final examination, you will not pass the course.

1. p ∨ q ∨ r

You either have the flu, miss the final examination or pass the course.

1. (p → ï¿¢r) ∨ (q → ï¿¢r)
   1. If you have the flu, you will not pass the course OR If you miss the final examination, you will not pass the course.
   2. You will not pass the course if you have the flu OR you miss the final examination.
2. (p ∧ q) ∨ (ï¿¢q ∧ r)

You have flu and you will miss the exam or you do not miss the exam you will pass the course.

Q2.    State the converse, contrapositive and inverse of each of the conditional statement. Answer:

1. If it snows tonight, then I will stay at home.

Inverse

      If it does not snow tonight, then I will not stay at home.

Converse

If I stay at home, then it snows tonight.

Contrapositive

If I don’t stay at home then it won’t be snowing tonight.

1. I go to the beach whenever it is a sunny summer day.

Inverse

            I don’t go to the beach whenever it is not a sunny summer day.

Converse

            Whenever I go to the beach, it is a sunny summer day.

Contrapositive

            If it is not a sunny summer day then I will not go to the beach.

1. When I stay up late, it is necessary that I sleep until noon.

Inverse

            If I do not stay up late, it is necessary that I don’t sleep until noon.

Converse

            If I sleep until noon, then I stayed up late.

Contrapositive

            If I don’t sleep until noon, then I don’t need to stay up late.

1. If it snows today, I will ski tomorrow.

Inverse

                        If it does not snow today, then I will not ski tomorrow.

Converse

                        If I ski tomorrow, it will snow today.

Contrapositive

            If I don’t ski tomorrow, it will not snow today. 

1. I come to class whenever there is going to be a quiz.

Inverse

            I don’t come to class if there is not going to be a quiz.

Converse

            If there is going to be a quiz then I will come to class.

Contrapositive

            If there is not going to be a quiz then will not come to the class.

1. A positive integer is a prime only if it has no divisors other than 1 and itself.

Inverse

            If a positive integer is not prime, it has divisors other than 1 and itself.

Converse

            If a positive integer has no divisors other than 1 and itself, it is prime.

Contrapositive

            If a positive integer has divisors other than 1 and itself, it is not prime.

  Q3.    Is the assertion “This statement is false” a proposition? Motivate your answer.

Answer.          The Assertion “This statement is false” is not a proposition because it is an incomplete description which has no reference.

Q4.    Use De Morgan’s Law to find the negation of each of the following statements.

1. Samir will take a job in industry or go to graduate school.
2. Ahmad knows java and calculus.
3. Samad is young and strong.
4. Aneela will move to Lahore or Gujrat.

Answer          

1. Samir will not take a job in industry and not go to graduate school.
2. Ahmad does not know java or calculus.
3. Samad is not young or not strong.
4. Aneela not move to Lahore or not move to Gujrat.

Q5.    Show that (p → q) ∧ (q → r) → (p → r) is tautology. Solution.

p	q	p → q
0	0	1
0	1	1
1	0	0
1	1	1

q	r	q → r
0	0	1
0	1	1
1	0	0
1	1	1

p	r	p → r
0	0	1
0	1	1
1	0	0
1	1	1

    

p → q	q → r	(p → q) ∧ (q → r)
1	1	1
1	1	1
0	0	0
1	1	1

(p → q) ∧ (q → r)	p → r	(p → q) ∧ (q → r) → (p → r)
1	1	1
1	1	1
0	0	1
1	1	1

 

 Thus (p → q) ∧ (q → r) → (p → r) is a Tautology.

 

 

 

 

 

Q6.    Show that (p ∨ q) ∧ (ï¿¢p ∨ r) → (q ∨ r) is tautology. Solution.

p	q	p ∨ q
0	0	0
0	1	1
1	0	1
1	1	1

p	r	ï¿¢p	ï¿¢p ∨ r
0	0	1	1
0	1	1	1
1	0	0	0
1	1	0	1

  

p ∨ q	ï¿¢p ∨ r	(p ∨ q) ∧ (ï¿¢p ∨ r)
0	1	0
1	1	1
1	0	0
1	1	1

                                                                        

 

 

 

 

q	r	q ∨ r
0	0	0
0	1	1
1	0	1
1	1	1

 

 

 

 

 

 

(p ∨ q) ∧ (ï¿¢p ∨ r)	q ∨ r	(p ∨ q) ∧ (ï¿¢p ∨ r) → (q ∨ r)
0	0	1
1	1	1
0	1	1
1	1	1

 

 

 

 

 

 

Thus (p ∨ q) ∧ (ï¿¢p ∨ r) → (q ∨ r) is tautology.

Q7.    Let N(x) be the statement “x has visited north”, where the domain consist of the students in UOH. Express each of these quantifications in English.

a) ∃xN(x)                                              b) ∀xN(x)                                             c) ¬∃xN(x)

d) ∃x¬N(x)                                          e)¬∀xN(x)                                           f) ∀x¬N(x)

Answer.

1. Some students has visited north.
2. All students in UOH has visited north.
3. None of student in UOH has visited north.
4. Some students in UOH has not visited north.
5. Some students in UOH has visited north.
6. All students in UOH has not visited north.

 

 

Q8.    Show that (using law) the following propositions are logically equivalent.

1. p ↔ q and (p ∧ q) ∨ (¬p ∧¬q)
2. p→ q∧p→r and p→(q∧r)

Solution.

1. p ↔ q and (p ∧ q) ∨ (¬p ∧¬q)

                        p ↔ q   ≡ (p → q) ∧ (q → p)

≡ (¬p ∨ q) ∧ (¬q ∨ p)

≡ ((¬p ∨ q) ∧ ¬q) ∨ ((¬p ∨ q) ∧ p), Distributive law

≡ ((¬p ∧ ¬q) ∨ (q ∧ ¬q)) ∨ ((¬p ∧ p) ∨ (q ∧ p)), Distributive law (twice)

≡ ((¬p ∧ ¬q) ∨ F) ∨ (F ∨ (q ∧ p)), Negation law (twice)

≡ (¬p ∧ ¬q) ∨ (q ∧ p), Identity law (twice)

≡ (p ∧ q) ∨ (¬p ∧ ¬q), Commutative law (twice)

 

1. p→ q∧p→r and p→(q∧r)

(p → q) ∧ (p → r)   ≡  (¬p ∧ q) ∧ (¬p ∧ r) Logical equivalence using conditionals

≡ (¬p ∧ ¬p) ∧ (q ∧ r) Associative and Commutative Laws

≡ ¬p ∧ (q ∧ r) Idempotent law

≡ p → (q ∧  r) Logical equivalence using conditionals.