Three Sets A, B and C are: A = {1, 2, 8,9,12,15,17}, B = { 1,2, 3 ,4, 10 } and C {7, 9, 10, 11, 13}. Find A U B A C ; A A ~B A U U C; A A B U C and (A A ~C).

Explain principle of multiplication with an example.

How many ways are there to distribute 15 district item into 5 distinct boxes with:

 i) At least two empty box.

 ii) No empty box

Explain addition theorem in probability.

A and B are mutually exclusive events such that P(A) = 1/4 and P(B) = 2/5 and P(AU B) = 1/2. What is the probability of P(A Ո B)

How many different professionals committees of 10 people can be formed, each containing at least 2 Professors, at least 3 Managers and 3 ICT Experts from list of 10 Professors, 6 Managers and 8 ICT Experts?

Explain principal of duality with the helpof example.

Prove that 2ⁿ > n³, ∀n ≥ 10

Show that:

           Q => P and (~Q P) are equivalent.

How many words can be formed using letter of PEPSUDENT using each letter at most once?

i) If each letter must be used,

ii) If some or all the letters may be omitted

What is a tautology? If P and Q are statements, show whether the statement

(P → Q) ∨ ( → P) is a tautologyor not.

Make logic circuit for the following Boolean expressions:

i) (x'y'z') + (x'yz)' + (xz'y)

ii) ( x'yz') (xyz') (x'y'z)

Explain inclusion-exclusion principle with example.

Show whether √5 is rational or irrational

Write down suitable mathematical statement that can be represented by the following symbolic properties.

i) ( ∃ x) ( ∃ y) ( ∀ z)P

ii) ∀ (z) ( ∃ y) ( ∃ z)Q

Draw a Venn diagram to represent followings:

i) (A   ⋂ B ⋂ C) (A ⋂ B ⋂ C)

ii) (A ⋂ B ⋂ C) (B ⋂ C)

Draw a Venn diagram to represent followings:

i) (A   ⋂ B ⋂ C) (A ⋂ B ⋂ C)

ii) (A ⋂ B ⋂ C) (B ⋂ C)

Give geometric representation for followings:

i) {5, 5) x ( -2, -2)

ii) {1, 5) x ( -2, -3)

Make truth table for followings.

 i) p→(~ ˅  ˅q) ˅ (~p ˅ r)

 ii) p→ (~r  ˅~ q) (p  ˅~ r)

State and explain De Morgan’s law for Boolean algebra