Using the truth table, show that: p q n q p)
P n q
Prove that sqrt(2) is irrational.
Find the Boolean expression for the output of the following circuit <img src='./qimages/9781-1c.jpg'> Make Venn diagram for the following set of expressions A A B (Symmetric difference)

(iii) A n B n C
A U B C
Let there be a relation f defined as
f Is f a function? If not, why How many distinct three-letter words can be formed from the letters of the word MAST? In how many ways can a student choose 8 questions out of 10 in an exam? A coin is tossed n times. What is the probability of getting exactly r heads?

2.(a) Prove the following:

<br><br> <img src='./qimages/9781-2a.jpg'> Use mathematical induction to prove that 1^2 2^2 3^2 ... n^2 n(n (2n 6
Write the contrapositive and converse of the following sentence
"If 2 2 then I am Prime Minister of India."
Explain proof by contradiction, with the help of an example.

3.(a) Reduce the following equations to simpler form: <img src='./qimages/9781-3a.jpg'> Const1"4ct logic circuits for the following Boolean expressions <img src='./qimages/9781-3b.jpg'> What is dual of a Boolean expression Explain the principle of duality with the help of an example.

4.(a) Describe the following region using intersection and union: <img src='./qimages/9781-4a.jpg'> Let A be a set and a relation R is defined on A such that aRb if a b. Check if R is reflexive

(ii) symmetric

(iii) transitive

(iv) asymmetric.
Let there be a function f A where A and B are sets defined as follows :

A B s}

f
Explain if f is
one to one
onto

(iii) bijective.

(d) Prove that A A n B using Venn Diagram.

5.(a) Make Pascal's triangle up to n 6. Let A and B be two mutually exclusive events such that 0·6 and 0·3. What is the probability that
A does not occur
A and B both occur simultaneously?
How many ways are there to distribute r distinct objects into 5 distinct boxes with no empty box Disprove the following statement:

<br><br> <img src='./qimages/9781-5d.jpg'>