Let f and g be the functions defined by $ f(x) = 2x+3 $ and $ g(x) = 3x-2 $ then composition of f and g is

Among 200 people, 150 either swim or jog or both. If 85 swim and 60 swim and jog, how many jog?

A graph in which all nodes are of equal degree is known as graph.

The minimum number of spanning trees in a connected graph with n nodes is

If a set contains exactly m distinct elements where m denotes some non-negative integer, then the set is

In an unweighted, undirected-connected graph, the shortest path from a node S to every other node is computed most efficiently, in terms of time complexity, by

The negation of 'Today is Friday' is

Whether the relation R on the set of all integers is reflexive, symmetric, anti-symmetric, or transitive, where $ (x, y) \in R $ if and only if $ xy \ge 1 $?

If $ p = \text{'It is raining'} $ and $ q = \text{'She will go to college'} $, 'It is raining and she will not go to college' will be denoted by

If $ x > 0 $ and $ x^2 < 0 $, then $ x \ge 10 $.

Define the following terms and give an example for each:
Reflexive relation, irreflexive relation, antisymmetric relation, partition set

Prove that, if $ S = \{s_1, s_2, \cdots, s_k\} $ be a set, then $ P(S) $ has $ 2^k $ elements.

Define 'group', 'order of a group', and 'Abelian group'. Prove that every subgroup of a cyclic group is cyclic.

Define the following with example:
Ring, homomorphism, cyclic group, coset

Determine whether f is one-one or onto for the following cases:
(i) Let $ A = B = \{1, 2, 3, 4\} $ and $ f = \{(1, 1), (2, 3), (4, 2)\} $
(ii) Let $ A = \{a, b, c\} $, $ B = \{1, 2, 3, 4\} $ and $ f = \{(a, 1), (b, 1), (c, 4)\} $

Suppose H and K are normal subgroups of G with $ H \cap K = \{1\} $. Show that $ xy = yx $ for all $ x \in H $ and $ y \in K $. Let $ I \in R $ be an ideal.

The radical $ \sqrt{I} = \{r \in R | rn \in I, n \in N\} $. Show that $ \sqrt{I} $ is an ideal.

State and prove De Morgan's laws of set theory.

In a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94 had taken a computer science course, 58 had taken a business course, 28 had taken both a mathematics and a business course, 26 had taken both a mathematics and a computer science course, 22 had taken both a computer science and a business course, and 14 had taken all three types of courses.
(i) How many of these students had taken none of the three courses?
(ii) How many had taken only a computer science course?

Prove that for any non-empty binary tree T, if $ n_0 $ is the number of leaves and $ n_2 $ be the number of nodes having degree two, then $ n_0 = n_2 + 1 $.

Derive total number of nodes of a binary tree having depth n.

What is graph? Differentiate between directed and undirected graph.

How many different ways can you represent a graph? Explain each of the representations by examples. What is complete graph?

Prove that the sum of degrees of all vertices in a graph is always even.

Consider the graph given below:
(a) Find the adjacency list and BFS traversal of the above graph.

Prove that the maximum number of edges possible in a simple graph of n nodes is $ n(n-1)/2 $.

Let f and g be the functions defined by $ f(x) = 2x+3 $ and $ g(x) = 3x-2 $ then composition of f and g is

Among 200 people, 150 either swim or jog or both. If 85 swim and 60 swim and jog, how many jog?

A graph in which all nodes are of equal degree is known as graph.

The minimum number of spanning trees in a connected graph with n nodes is

If a set contains exactly m distinct elements where m denotes some non-negative integer, then the set is

In an unweighted, undirected-connected graph, the shortest path from a node S to every other node is computed most efficiently, in terms of time complexity, by

The negation of 'Today is Friday' is

Whether the relation R on the set of all integers is reflexive, symmetric, anti-symmetric, or transitive, where $ (x, y) \in R $ if and only if $ xy \ge 1 $?

If $ p = \text{'It is raining'} $ and $ q = \text{'She will go to college'} $, 'It is raining and she will not go to college' will be denoted by

If $ x > 0 $ and $ x^2 < 0 $, then $ x \ge 10 $.

Define the following terms and give an example for each:
Reflexive relation, irreflexive relation, antisymmetric relation, partition set

Prove that, if $ S = \{s_1, s_2, \cdots, s_k\} $ be a set, then $ P(S) $ has $ 2^k $ elements.

Define 'group', 'order of a group', and 'Abelian group'. Prove that every subgroup of a cyclic group is cyclic.

Define the following with example:
Ring, homomorphism, cyclic group, coset

Determine whether f is one-one or onto for the following cases:
(i) Let $ A = B = \{1, 2, 3, 4\} $ and $ f = \{(1, 1), (2, 3), (4, 2)\} $
(ii) Let $ A = \{a, b, c\} $, $ B = \{1, 2, 3, 4\} $ and $ f = \{(a, 1), (b, 1), (c, 4)\} $

Suppose H and K are normal subgroups of G with $ H \cap K = \{1\} $. Show that $ xy = yx $ for all $ x \in H $ and $ y \in K $. Let $ I \in R $ be an ideal.

The radical $ \sqrt{I} = \{r \in R | rn \in I, n \in N\} $. Show that $ \sqrt{I} $ is an ideal.

State and prove De Morgan's laws of set theory.

In a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94 had taken a computer science course, 58 had taken a business course, 28 had taken both a mathematics and a business course, 26 had taken both a mathematics and a computer science course, 22 had taken both a computer science and a business course, and 14 had taken all three types of courses.
(i) How many of these students had taken none of the three courses?
(ii) How many had taken only a computer science course?

Prove that for any non-empty binary tree T, if $ n_0 $ is the number of leaves and $ n_2 $ be the number of nodes having degree two, then $ n_0 = n_2 + 1 $.

Derive total number of nodes of a binary tree having depth n.

What is graph? Differentiate between directed and undirected graph.

How many different ways can you represent a graph? Explain each of the representations by examples. What is complete graph?

Prove that the sum of degrees of all vertices in a graph is always even.

Consider the graph given below:
(a) Find the adjacency list and BFS traversal of the above graph.

Prove that the maximum number of edges possible in a simple graph of n nodes is $ n(n-1)/2 $.