LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600034
SECOND CONTINUOUS INTERNAL ASSESSMENT - UG
DEPARTMENT OF MATHEMATICS
Date: 27.01.2022
Course Code: MT6652a
Max Marks: 40
Time: 08.30 – 10.00am
APPLIED ALGEBRA
PART A
Answer All Questions:
1. Define Partial ordered relation.
2. When do you say a mapping to be order preserving?
3. List the conditions for a mapping to be a lattice homomorphism.
4. What is the sub-Boolean algebra?
5. State the condition for a Boolean expression to be symmetric.
(5 x 2 = 10)
PART B
(2 x 5 = 10)
Answer ANY TWO questions:
6. State and prove any two properties of Lattices.
7. Prove that any lattice homomorphism is order preserving,
8. Write the following Boolean expressions in an equivalent sum-of-products canonical forms
in three variables 𝑥1, 𝑥2 and 𝑥3:
(i)
𝑥1 * 𝑥2
(ii)
𝑥1⊕𝑥2
(iii)
(𝑥1 ⊕ 𝑥2)' * 𝑥3
9. In a complemented distributive lattice, show that the following are equivalent:
'
'
'
'
𝑎≤𝑏⇔𝑎∧𝑏 = 0⇔𝑎 ∨𝑏 = 1⇔𝑏 ≤ 𝑎 .
PART C
Answer ANY ONE Question:
(1 x 20 =20)
10. Reduce the following expressions:
'
a. 𝑎∙𝑎 𝑏
b. 𝑎(𝑎 + 𝑐) = 𝑎𝑎 + 𝑎𝑐
'
'
c. 𝑎𝑏 + 𝑎𝑏𝑐 + 𝑎𝑏𝑐 + 𝑎 𝑏𝑐
'
'
'
d. 𝑎 𝑏𝑐 𝑑 + 𝑎 𝑏𝑐𝑑 + 𝑎𝑏𝑑
11. Prove that 𝑆
36
the set of all divisors of and D denote the relation of division is a lattice. Also
evaluate the diagrams for 𝑆𝑛; 𝑛 = 12, 8