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MTH202 Discrete Mathematics Assignment 1 Solution Spring 2013

Q1: Which of the following are the propositions? Write Truth value of each proposition.
1. Ali is an intelligent boy.

2. She cooks well.

3. 5 + 6 = 12

4. x + 5 > 20

5. The sun sets in the west.

6. Complete your homework.

7. Respect your elders.

8. The set of natural numbers begins with 1.

9. The square root of 2 is irrational.

10. The sky is blue at night.

11. Honesty is the best policy.

Solution:

1. Ali is an intelligent boy.   ( T )
2. She cooks well.      ( T )
3. 5 + 6 = 12       ( T )
4. x + 5 > 20        ( F )
5. The sun sets in the west.    ( T )
6. Complete your homework.    ( F )
7. Respect your elders.         ( F )
8. The set of natural numbers begins with 1.    ( T )
9. The square root of 2 is irrational.         ( T )
10. The sky is blue at night.          ( T )
11. Honesty is the best policy.      ( T )
Q2: Let p = I am a student of Computer Science. q = I work hard. Then translate the following logical expressions into English sentences.

( ) Use the result: ~ ( p q ) = p ~ q
p q qp pq pq qp pq → → ↔ → → → → ∧ ∼∼ ∼∼ ∼
Q3: Let h = Asad is happy. s = Asad is sad. t = Asad watch television. Then translate the following logical expressions into English sentences.

hts st th t h s t h s →∧ ↔ → →∨ ∧↔ ∼ ∼ ∼∼ ∼ ∼

h=asad is happy

s=asad is sad

t=asad watch tv

Solution

1;h impliz t hat tilda s=asad is happy if he watch tv and he is not sad

2;s bicondishnal tilda t= asad is sad if and only if he dose not watch tv

3;tilda t impliz tilda h= asad is not watching tv if he is not happy.

4;tilda impliz h vel s=asad is not watching tv if he is happy or sad

5;t hat h bicondishnal tilda s= asad watch tv and he is happy if and only if he is not sad..

 
Q4: Write converse, inverse, contra-positive of the following conditional sentences: a. “If Ali plays soccer, then Bilal plays hockey.” b. “If it is spring season, then trees are green.” c. “If x is a natural number, then 2x is an even number.”

Solution:

CONVERSE:

A: “If Ali plays soccer, then Bilal plays hockey.

ANS: If bilal plays hockey, then ali plays soccer.

b. “If it is spring season, then trees are green.”

ANS: If trees are green then it is spring season.

C. If x is a natural number, then 2x is an even number.”

ANS: If 2x is an even number then x is a natural number.

INVERSE:

A: “If Ali plays soccer, then Bilal plays hockey.

ANS: If ali does not play soccer then bilal does not play hockey.

b. “If it is spring season, then trees are green.”

ANS: If it is not spring then trees are not green.

C. If x is a natural number, then 2x is an even number.”

ANS: If x is  not a natural number then 2x is not an even number.

 

CONTRA-POSITIVE:

A: “If Ali plays soccer, then Bilal plays hockey.

ANS: If bilal does not play hockey then ali does not play soccer.

b. “If it is spring season, then trees are green.”

ANS: If trees are not green then it is not spring season.

C. If x is a natural number, then 2x is an even number.”

ANS: If 2x is not an even number then x is not a natural number.

 
Q5: Construct the Truth table for () p q →∼ ( Hint: Use the result: () p qpq → ≡ ∧ ∼∼ )

truth table

p   q   p impliz q  tilda(p impliz q)

t     t       t                 f

t     f        f                 t

f      t       t                 f

f       f       t               f

 
Q6: What is the Truth value of each of the following logical expressions? Given : p = True, q = False
( For example: p q = true false = true )
() ( ) ( p q ) ( ) p q ( ) p q ( ) q p ( ) q ~ p ( ) p q a p q b c d e f g
∨∨
⊕ ∧ ∨

 Solution:

1; p+ =t

2:tilda p hat q = f

3: tilda p vel tilda q= t

4;p bicndishnal q= f

5:p impliz q= f

6:tilda q impliz p= t

7: tilda p impliz tilda q= t

 

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