Q.1) If $A = \{2, 4, 6, 9\}$ and $B = \{4, 6, 18, 27, 54\}$, $a \in A$, $b \in B$, find the set of ordered pairs such that 'a' is factor of 'b' and $a < b$. (June-2023)

Q.2) Let R be the relation on the set R of all real numbers defined by a R b if and only if $|a - b| \le 1$ Then prove that R is reflexive, symmetric, but not transitive. (June-2023)

Q.3) Let $A = \{1, 2, 3, 4, 5, 6\}$ and a Rb if and only if a is multiple of b.
Find:
i) Domain
ii) Range,
iii) Matrix of a relation,
iv) Digraph of the relation R. (June-2023)

Q.4) What common relations on Z are the transitive closures of the following relations?
i) a Sb if and only if $a + 1 = b$,
ii) a Rb if and only if $a - b = 2$. (June-2023)

Q.5) Let $A = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24\}$ be ordered by divisibility. Draw Hasse diagram. (June-2023)

Q.6) Write short note on partially ordered sets and explain with suitable example. (June-2023)

Q.7) Write short Note on: Hasse diagram (June-2023)

Q.8) Write short Note on: Lattice (June-2023)

Q.9) Let T be the set of positive integers less than 10 and let "R be the relation on T defined by aRb if and only if $|a^2 - b^2|$ is a multiple of 5". Prove that R is reflexive, symmetric, but not transitive. (June-2024)

Q.10) Consider a set C = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15} ordered by divisibility. Draw its Hasse diagram. (June-2024)

Q.11) Consider two sets A and B. Let $A = \{2, 3, 4, 5\}$ and $B = \{4, 5, 6, 7\}$ Using Venn diagrams, prove the following statements:
Prove that $(A \cup B) = A' \cap B'$.
Prove that $(A \cap B)' = A' \cup B'$. (June-2024)

Q.12) Let A be the set of even numbers between 10 and 20, and B be the set of multiples of 5 between 15 and 35. If $a$ is an even number from set A, and $b$ is a multiple of 5 from set B, find the set of ordered pairs such that '$a$' is a factor of '$b$' and $a$ is less than $b$. (Dec-2024)

Q.13) Consider the relation R on the set of integers Z defined by aRb if and only if $a^2 - b^2$ is divisible by 4. Determine the common relations on Z that are the transitive closures of R. (Dec-2024)

Q.14) Suppose that the relation R is irreflexive. Is $R^2$ necessarily irreflexive? Give a reason for your answer. (Dec-2024)

Q.15) Prove that:
i) $A - (B \cap C) = (A - B) \cup (A - C)$
ii) $A \times (B \cap C) = (A \times B) \cap (A \times C)$ (Nov-2023)

Q.16) With the help of Venn-diagram, prove that
$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$ (Nov-2023)

Q.17) Consider the set Z of integers m > 1. We say that $x$ is congruent to $y$ modulo $m$ written as $x \equiv y \pmod m$
If $x - y$ is divisible by $m$. Show that this defines an equivalence relation on Z. (Nov-2023)

Q.18) Let $D_m$ denotes the positive divisors of $m$ ordered by divisibility. Draw the Hasse diagrams of the following:
i) $D_{15}$
ii) $D_{24}$ (Nov-2023)

Q.1) Given the recurrence relation $a_n = a_{n-1} + 4a_{n-2}$ with initial conditions $a_0 = 2$ and $a_1 = 5$, determine the closed-form expression for the sequence. (June-2024)

Q.2) Show that the set $H = \{e, a, a^2, a^3\}$ is a subgroup of a group G where $G = \{e, a, a^2, a^3, b, ab, a^2b, a^3b\}$. (June-2024)

Q.3) Let (A, *) be a semigroup. Show that, for a, b, c in A, if $a * c = c * a$ and $b * c = c * b$, then $(a * b) * c = c * (a * b)$. (June-2024)

Q.4) Write short note on: Recurrence Relation (June-2024)

Q.5) Solve the recurrence relation $a_n = a_{n-1} + 6a_{n-2}$ given the initial conditions $a_0 = 3$ and $a_1 = 6$. (Dec-2024)

Q.6) Prove that a subgroup H of a group G is normal if and only if $xHx^{-1} = H, \forall x \in G$. (Dec-2024)

Q.7) Let G = {0, 1, 2, 3, 4, 5} with addition modulo 6. Is G a group? Justify your answer. (Dec-2024)

Q.8) Write short note on: Abelian group and cyclic group (Dec-2024)

Q.9) Consider the set Q of rational numbers and let * be the operation on Q defined by:
$a * b = a + b - ab$
i) Is (Q, *) a semi-group. Is it commutative.
ii) Find the identity element for *.
iii) Do any elements in Q have inverse? What is it? (Nov-2023)

Q.10) Define Ring with example. Also explain Commutative ring and ring homomorphism. (Nov-2023)

Q.11) Solve the following recurrence relations
$a_n = 3a_{n-1} - 3a_{n-2} + a_{n-3}, a_0 = 0, a_3 = 3, a_5 = 10$ (Nov-2023)

Q.1) i) Prove that $p \wedge q \Rightarrow q \vee p$ is a Tautology.
ii) Show that $(p \vee q) \wedge (-p) \wedge (-q)$ is a contradiction (June-2023)

Q.2) Prove that in any graph G, even number of vertices is of odd degree. (June-2023)

Q.3) Let G be a planar graph with 10 vertices, 3 components and 9 edges. Find the number of regions in G (June-2023)

Q.5) Write short Note on: Weighted Graph (June-2023)

Q.6) Let G be a simple graph. Show that the relation R on the set of vertices of G such that uRv if and only if there is an edge associated to (u, v) is a symmetric, irreflexive relation on G. (June-2024)

Q.9) Find a least expensive route in terms of total tolls using the roads in the graph between the pairs of cities in part (A). (June-2024)

Q.10) Write short note on: Tautologies and Contradiction (June-2024)

Q.11) Construct a niche overlap graph for six species of birds, where the hermit thrush competes with the robin and with the blue jay, the robin also competes with the mockingbird, the mockingbird also competes with the blue jay, and the nuthatch competes with the hairy wood-pecker. (Dec-2024)

Q.14) Show that
$((p \lor q) \land \sim(\sim p \land (\sim q \lor \sim r))) \lor (\sim p \land \sim q) \lor (\sim p \land \sim r) \equiv \text{T}$
is a tautology by laws of algebra of propositions. (Nov-2023)

Q.15) Obtain the conjunctive normal form of
i) $p \land (p \Rightarrow q)$
ii) $\sim p \Rightarrow [r \land (p \Rightarrow q)]$ (Nov-2023)

Q.17) What is coloring problem? Hence define coloring of graph. (Nov-2023)

Q.1) Solve the equations
$6x + 15y + 55z = 76$,
$15x + 55y + 225z = 295$,
$55x + 225y + 979z = 1259$
Using Cholesky decomposition method. (June-2023)

Q.2) Find singular value decomposition composition for for $A = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}.$ (June-2023)

Q.3) Calculate the gradient of the function $f(X) = Tr(X^T X)$ where X is a matrix. (June-2024)

Q.4) Write a short note on Cholesky Decomposition. (June-2024)

Q.5) Given a matrix F with SVD $F = U\Sigma V^T$, where:
$$U = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \Sigma = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}, V = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
Find the original matrix F. (Dec-2024)

Q.6) Find the Determinant and trace of the matrix:
$$\begin{pmatrix} 2 & -1 & 0 \\ 3 & 4 & 2 \\ 1 & 0 & -3 \end{pmatrix}$$ (Dec-2024)

Q.7) Using Cholesky decomposition, Verify that whether the following matrix is positive definite or not. (Dec-2024)

Q.8) Write short note on: Eigen decomposition. (Dec-2024)

Q.9) Solve the simultaneous equations:
$25x + 15y - 5z = 35$
$15x + 18y + 0.z = 33$
$-5x + 0.y + 11z = 6$
Using Cholesky Decomposition. (Nov-2023)

Q.10) Find the singular value decomposition of the matrix:
$$A = \begin{bmatrix} -4 & -7 \\ 1 & 4 \end{bmatrix}$$ (Nov-2023)

Q.1) To test the hypothesis that eating fish makes one smarter, a random sample of 12 persons take a fish oil supplement for one year and then are given an IQ test. Here are the results:
116   111   101   120   99   94   106   115   107   101   110   92
Test using the following hypothesis, report the test statistic with the P-value, then summarize your conclusion.
$H_0: \mu = 100$
$H_a: \mu > 100$ (June-2023)

Q.2) What are the critical values for a one-independent sample non directional (two-tailed) z test at a .05 level of significance? (June-2023)

Q.3) Define Hypothesis. Write a short note on Null Hypothesis and Alternative Hypothesis. (June-2024)

Q.4) Suppose the average salary of a principal in a college in the private sector is Rs. 45,000 per month, and a sample of 50 principals has a mean of Rs. 48,326 with a significance level of $\alpha = 0.05$. Can we claim that principals earn Rs. 45,000 per month, given that the standard deviation of the population is Rs. 1,523? (June-2024)

Q.5) Write short note on: Time Series Analysis (June-2024)

Q.6) Write a detail note on Analysis of Variance. (Dec-2024)

Q.7) A sample of 100 B.Sc. Student is taken from a medical college with height having standard deviation (S.D.= 10 cm). The mean height of sample of student is found 168.8 cm. Can we accept that the mean height of student is 170 cm. Justify your answer. (Dec-2024)

Q.8) Write short note on: Type-I and Type-II Errors. (Dec-2024)

Q.9) What is null hypothesis? What is its significance in statistical variance? (Nov-2023)

Q.11) A coin was tossed 400 times and the head turned up 216 times. Test the hypothesis that the coin is unbiased. (Nov-2023)