TEAM 1  -

 Assignment

1. Determine whether each of the following sentences is a statement.

a) In 1998 William Clinton was the  President  of  the  United  states.  - It is a statement.

b) x+3 is a positive integer.-    Not a statement.

c) Fifteen is an even number. -   It is a statement.

d) If Jennifer is late for the party, then her cousin Zachary will be quite angry.- It is a statement.

 e) What time is it?     -Not a statement.

f)  As  of  June 30,1998,Christine Marie Evert had  won the French Open seven times.- It is a statement

2) Use truth tables to verify the following logical equivalences

i) p → (q/\r)<==>(p→ q)/\(q→ r)

ii) [(p\/q) → r ] <==>[ ( p→ r ) /\(q→ r)]

iii) [p→ (q\/r)] <==> [¬r→ (p→ q)]

Ans : i)

P	Q	r	q/\r	p→(q/\r)	p→ q	p→ r	(p→ q)/\(p→ r)
0	0	0	0	1	1	1	1
0	0	1	0	1	1	1	1
0	1	0	0	1	1	1	1
0	1	1	1	1	1	1	1
1	0	0	0	0	0	0	0
1	0	1	0	0	0	1	0
1	1	0	0	0	1	0	0
1	1	1	1	1	1	1	1

Hence they are logically equivalent.

 Ans : ii)       

P	Q	r	p\/q	(p\/q)→r	p→ r	q→ r	(p→ r)/\(q→ r)
0	0	0	0	1	1	1	1
0	0	1	0	1	1	1	1
0	1	0	1	0	1	0	0
0	1	1	1	1	1	1	1
1	0	0	1	0	0	1	0
1	0	1	1	1	1	1	1
1	1	0	1	0	0	0	0
1	1	1	1	1	1	1	1

    Hence they are logically equivalent.

Ans : iii)

P	Q	r	q\/r	p→(q\/r)	p→ q	¬r→(p→ q)
0	0	0	0	1	1	1
0	0	1	1	1	1	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	0	0	0	0
1	0	1	1	1	0	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

Hence they are logically equivalent.

 b). Use the substitution rules to show that

Ans :  [p→ (q\/r)]<=> [(p/\¬q) → r]

          [p→ (q\/r)] ó [¬r→ (p→ q)]                       From part (iii) of part (a)

                               ó [¬r→ (¬p\/q)]                      By the 2^nd Substitution Rule,

                                                                                         And (p→ q)ó(¬p\/q)

                                 ó [¬ (¬p\/q)→  ¬¬r]                By the 1^st Substitution Rule,

          And (s→ t)ó (¬t → ¬s) for any

            Primitive statement   s, t

                             ó [(¬¬p/\¬q) → r]                      By De Morgan’s   Law, Double Negation

                             ó [(p /\¬q) → r]         By Double Negation and   2^nd Substitution Rule

3. The following are three valid arguments. Establish the validity of each by means of a truth table.In each case, determine   which rows of the table are crucial for assessing the validity of the argument and which rows can be ignored.

a)      [p/\(p→ q)/\r]→[(p\/q)→r]

b)      [[(p/\q)→r]/\¬q/\(p→¬r)]→(¬p\/¬q)

c)       [[p\/(q\/r)/\¬q]→(p\/r)

Ans:  a)

P	q	R	p→q	p\/q	(p\/q)→r
0	0	0	1	0	1
0	0	1	1	0	1
0	1	0	1	1	0
0	1	1	1	1	1
1	0	0	0	1	0
1	0	1	0	1	1
1	1	0	1	1	0
1	1	1	1	1	1

The validity of the argument follows from the results in the last row.

Ans: b)

p	q	R	p/\q	(p/\q)→r	¬q	¬r	p→¬r	¬p\/¬q
0	0	0	0	1	1	1	1	1
0	0	1	0	1	1	0	1	1
0	1	0	0	1	0	1	1	1
0	1	1	0	1	0	0	1	1
1	0	0	0	1	1	1	1	1
1	0	1	0	1	1	0	0	1
1	1	0	1	0	0	1	1	0
1	1	1	1	1	0	0	0	0

The validity of the argument follows the results in 1, 2, 5.

Ans: c)

p	Q	R	q\/r	p\/(q\/r)	¬q	p\/r
0	0	0	0	0	1	0
0	0	1	1	1	1	1
0	1	0	1	1	0	0
0	1	1	1	1	0	1
1	0	0	0	1	1	1
1	0	1	1	1	1	1
1	1	0	1	1	0	1
1	1	1	1	1	0	1

The results in rows 2,5,6 establish the validity of the given argument.

4) Which of the following sets are equal?

a) { 1,2,3 }                               b) { 3,2,1,3 }

c) {3,1,2,3 }                             d) {1,2,2,3}

Ans : As all the sets contains same elements ,therefore they are all same set.