Discrete Mathematics Answers

Discuss two examples on binary trees both quantitatively and qualitatively.

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra

1. Develop truth tables and its corresponding Boolean equation for the following scenarios.
i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.''
ii. If it rains and you don't open your umbrella, then you will get wet.

1. Check whether the set is a group under the binary operation ‘*’defined as for any two elements .
2. i. State the relation between the order of a group and the number of binary operations that can be defined on that set.
ii. How many binary operations can be defined on a set with 4 elements?
3. Discuss the group theory concept behind the Rubik’s cube.

1. Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way.
2. i. State the Lagrange’s theorem of group theory.
ii. For a subgroup H of a group G, prove the Lagrange’s theorem.
iii. Discuss whether a group H with order 6 can be a subgroup of a group with order 13 or not. Clearly state the reasons.

1. Describe the characteristics of different binary operations that are performed on the same set.
2. Justify whether the given operations on relevant sets are binary operations or not.
i. Multiplication and Division on se of Natural numbers
ii. Subtraction and Addition on Set of Natural numbers
iii. Exponential operation: on Set of Natural numbers and set of Integers

Part 2
1. Write the multisets of prime factors for the given numbers.
I. 160
II. 120
III. 250
2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately.
3. Find the cardinalities of each multiset in part 2-1.

For the language L = {acnb | n > 0 }, which one of the following strings is not a part of the language ?

Use the symbols ~, ^ and v, and write the following statements.
both p or q and r

2. Produce truth tables for given Boolean expressions.

i. A̅ B̅ C+A B̅ C̅ + ABC +A̅ B C̅
ii. (A+B̅+C)(A+B+C)(A̅+B+C̅)