Assignment-1
For management extension (2016 B.C)

Attempt all questions in a group of at most 5 members. Show all the necessary steps clearly and neatly.

1. Let [tex]\( p \)[/tex] represent the statement: "Students are happy" and let [tex]\( q \)[/tex] represent the statement: "Teachers are happy." Translate the symbolic compound statement [tex]\( p \Rightarrow \neg q \)[/tex] into words.

2. For each of the following statements, identify if it is a proposition or not:
i. Mekelle is a city in India.
ii. [tex]\( x^2 + 1 = 0 \)[/tex]
iii. Every rectangle is a square.
iv. May God bless you!
v. Close the door.
vi. How old are you?
vii. There are some real numbers whose square is a negative real number.

3. Write the negation of each of the following sentences:
a) 7 is greater than 4 or 6 is less than 7.
b) It is raining and the weather is cold.
c) [tex]\( (\exists x \in \mathbb{R}) \left( |x| \ \textgreater \ x^2 \right) \)[/tex]
d) [tex]\( 2 + 3 \ \textgreater \ 4 \)[/tex]
e) There are some students who miss the final exam.

4. Translate to symbolic form and check the validity of the argument given by: If the rain does not come, then the crops are ruined and the people will starve. The crops are not ruined or the people will not starve. Therefore, the rain comes.

5. Determine the truth value of the following statements:
a) [tex]\( (\exists x \in \mathbb{R}) \left( x^2 - x = 0 \right) \)[/tex]
b) [tex]\( (\forall x \in \mathbb{N}) \left( x + 1 \geq 2 \right) \)[/tex]
c) [tex]\( (\forall x \in \mathbb{R}) \left( \sqrt{x^2} = x \right) \)[/tex]

6. For statements [tex]\( p \)[/tex] and [tex]\( q \)[/tex], show that the compound statement [tex]\( p \Rightarrow (p \vee q) \)[/tex] is a tautology.