Assignment no.01
Discretemathematics & probability (MTH-202)
solution
1:
Which of the following are the propositions? Write Truth value of each proposition.
1.Ali is an intelligent boy. (F) PREPOOSITION
2.She cooks well. NOT A PREPOSITION
3.5 + 6 = 12 (F) PREPOOSITION
4.x + 5 > NOT A PREPOOSITION
5.The sun sets in the west. (T) PREPOOSITION
6.Complete your homework. NOT A PREPOOSITION
7.Respect your elders. NOT A PREPOOSITION
8.The set of natural numbers begins with 1. (T) PREPOOSITION
9.The square root of 2 is irrational. (F) PREPOOSITION
10.The sky is blue at night. (F) PREPOOSITION
11.Honesty is the best policy. (T) PREPOOSITION
Q2:
Let p = “I am a student of Computer Science.”
q = “I work hard.”
Then translate the following logical expressions into English sentences.
a) p → q
If i am a student of Computer Science then I must work hard.
b) q → p
If I work hard then I must be Computer Science Student.
c) p ↔ q
If I am a student of Computer Science if and only if I work hard.
d) ~ p → ~ q
If I am not a student of Computer Science then I must not work hard.
e) ~ q → ~ p
if I m not working hard then I must not be a Student of Computer Science.
f) ~ (p →q) Use the result: ~ (p → q) = p ˅ ~ q
I m a student of Computer Science and I don’t work hard.
Q3:
let h = “Asad is happy.”
S = “Asad is sad.”
t = “Asad watch television.”
Then translate the following logical expressions into English sentences
a) h → t ˅ ~ s
if Asad is happy then he watch TV and he is not sad.
b) s ↔ ~ t
Asad is sad if and only if he is not watch TV.
c) ~ t → ~ h
If Asad is not watch TV then he is not happy.
d) ~ t → h ˄ s
Asad is not watch TV then he is happy or he is sad.
e) t ˄ h ↔ ~ s
Asad watch TV or 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.
Converse: If Bilal plays hockey, then Ali plays soccer.
Inverse: If Ali not plays soccer, then Bilal is not plays hockey
contra-positive If Bilal not plays hockey, then Ali is not plays soccer.
b: If it is spring season, then trees are green.
Converse: If trees are green, then its spring season.
Inverse: If it is not a spring season, then the trees are not green.
contra-positive If trees are not green, then its not spring season.
c: if x is a natural number, then 2x is an even number.
Converse: If 2x is an even number, then the x is a natural number.
Inverse: If x is not a natural number, then 2x is not an even number.
Contra-positive: If 2x is not an even number, then the x is not a natural number.
Q5:
Construct the truth table for ~ (p → q)
(Hint: Use the result : ~ ( p → q ) = ( p ˄ ~ q)
p | q | ~ q | p → q | ~ ( p → q ) | ( p ˄ ~ q) |
T | T | F | T | F | F |
T | F | T | F | T | T |
F | T | F | T | F | F |
F | F | T | T | F | 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 | p q | ~ ( p ˄ q ) | ~ q → ~ p | p → q | ~ q → p | ~ p ˅ ~ q | p ↔ q |
T | F | F | T | F | T | T | T | F | T | T | F |
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