Solved: There are 10 stalls for animals in an exhibition. Three animals; lion, pussycat and horse are to be exhibited. Animals of each kind are not less than 10 in number. What is the possible number of ways of arranging the exhibition?

Problem B Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

Q3 X people are chosen from a volley ball team (Take a value of X by yourself, possibly that number must be close to a number of volley ball team members).

Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) a) Figure out Euler graph from these three graphs. b…

Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2) a) Figure out Euler graph from these three…

Q3 X people are chosen from a volley ball team (Take a value of X by yourself, possibly that number must be close to a number of volley ball team members). (2+2+2) a) How many ways are there to choose Y people to take them to ground(T…

Q1 Draw any balance rooted tree having nodes from a to i. (2+2+2+2) Find the level of each vertex in the drawn tree. Find the height of this tree. Which vertices are descendants of node g.

(b) Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (i) R1 = {(a,b) : a divides b} (ii) R2 = {(a,b) : a ≤ b} (iii) R3 = {(a,b) : a 2 = b} (iv) R4 = {(a,b) : a + b ≤ 5} (v) R5 = {(a,b) : a = 2 and…

(a) Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. (i) R1 = {(2,x), (y,4), (6,z)} (ii) R2 = {(4,y), (y,4)} (iii) R3 = {(2,x), (4,y), (6,z)} (iv) R4 = {(4,y), (6,x), (4,x)} (v) R5 = {(x,2), (4…

(a) Let A = {0,1,2}. R = {(0,0), (0,1), (0,2), (1,1), (1,2), (2,2)} and S = {(0,0), (1,1), (2,2)} be two relations on A. (i) Show that R is a partial order relation. (ii) Is R a total order relation? (iii) Show that S is an equivalenc…

(b) (i) Let A = {2,4,6} and R = {(2,2), (2,4), (2,6), (4,4), (6,6)} be a relation on A. Find R−1 . (ii) Let A = {6,8,10,15} and B = {2,3,4}. Define a relation R from A to B by (x,y) ∈ R iff x-y is divisible by 2. Find R−1

3. (a) Let A = {1,2,3,4,5} and R = {(1,1), (1,3), (1,4), (2,2), (2,5), (3,1), (3,3), (3,4), (4,1), (4,3), (4,4), (5,2), (5,5)}. (i) Prove that R is an equivalence relation on A. (ii) Find the equivalence classes of 1 and 2. (b) Let A …

4. (a) Let R be the relation on Z × Z such that ((a,b), (c,d)) ∈ R ↔ a + d = b + c. Show that R is an equivalence relation. (b) If R is an equivalence relation on a finite non empty set A, then the equivalence classes of R all have t…

List the ordered pairs in the equivalence relations produced by these partitions of {a,b,c,d,e,f,g}. (a) {a,b}, {c,d}, {e,f,g} (b) {a}, {b}, {c,d}, {e,f}, {g} (c) {a,b,c,d}, {e,f,g} (d) {a,c,e,g}, {b,d}, {f}

Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2) a) Figure out Euler gr…

Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2) a) Figure out Euler graph from these three…

how to prove the following using defined sets A=B⇔A⊆B and B ⊆ A

Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (i) R1 = {(a,b) : a divides b}

Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (ii) R2 = {(a,b) : a ≤ b}

Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (iii) R3 = {(a,b) : a 2 = b}

Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (iv) R4 = {(a,b) : a + b ≤ 5}

Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A. (v) R5 = {(a,b) : a = 2 and a + 2b ≤ 10}

Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. (i) R1 = {(2,x), (y,4), (6,z)}

Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. (ii) R2 = {(4,y), (y,4)}

Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. (iii) R3 = {(2,x), (4,y), (6,z)}

Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. (iv) R4 = {(4,y), (6,x), (4,x)}

Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B. v) R5 = {(x,2), (4,z), (2,z), (6,y)}

Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2) a) Figure out Euler graph from these three g…

If A is the set of all residents of the United states, B the set of all Canadian citizens, and C the set of all women in the world, describe the sets A intersection B intersection B intersection C A-B A-C C-A

There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? b) In how many ways can …

A set is a fundamental building block in mathematics, and it is defined as a collection of well-defined objects.A set is always defined in a computer program with respect to an underlying universal set, and the elements in the univers…

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights.

Let A={0,1,2}. R={(0,0),(0,1),(0,2),(1,1),(1,2),(2,2)} and S={(0,0),(1,1),(2,2)} be two relations on A. 1.Show that R is a partial order relation 2.Is R a total order relation? 3.Show that S is an equivalence relation.

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights.

Take 5 nodes & 8 edges, Draw any Directed graph (Take help from book, but DO NOT copy paste any graph from examples or exercise.) a) Determine whether the given graph has Hamilton circuit, if it does, find such circuit. b) If not, pro…

Q3 In a prize distribution ceremony. There are first prize winner, 2nd prize winner, 3rd prize winner and fourth prize winner. Calculate the number of ways to select these winners from ___________ people who have been selected in this…

Q1 Find the floor and ceiling Function of x and . Take any value of x and y. (5) X Floor Ceiling -X.Y (x point y) X / Y (x divided by y)

Two fire sirens give alarms at intervals of 5/7, 7/8 hours. Since these two fire sirens sounded an alarm at 04:00 on Friday at the same time, on which day and at what time do they alarm together again?

13 and 65 are relatively prime? True or false

F(x)=x is one to one function true or false

Q4 Write down any 4*4 matrix having only zeros and ones. (3*5=15) a) Draw the Directed graph of that matrix. b) List the ordered pairs in the relation on set {1, 2, 3, 4} corresponding to this matrix. c) Determine whether the relatio…

a) 04 points Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions. AC B, CC B, AnC +0. (b) 03 points For each integer m, let Tm = {m2, m*}. How many elements are in each of T-3, T-1, To, T1? Give justificat…

Find formula for tha sequences with tha following first five terms 1,1/2,1/4,1/8,1/16

1. (a) Let A = {1,2,3,4} and B={0,3,6,8,12,15}. Consider a rule f(x) = x 2 − 1, x ∈ A. Is f be a function from A to B?

Let A = {a,b,c,d} and B = {c,d,e,f,g}. Let R1 = {(a,c), (b,d), (c,e)} R2 = {(a,c), (a,g), (b,d), (c,e), (d,f)} R3 = {(a,c), (b,d), (c,e), (d,f)} Justify which of the given relation is a function from A to B. (c) Let f be a real valued…

(a) (i) Let f : R → R be defined by the equation f(x) x 2 + 1. Let H ⊆ R and H = { y ∈ R : 5 ≤ y ≤ 10}. Then determine the inverse image f −1 (H).

Let g : R → R defined by the equation g(x) = x 2 + x. Let H ⊆ R and H = {y ∈ R : 6 ≤ y ≤ 12}. Then determine the inverse image g −1 (H).

(b) Determine whether each of these function is a bijection (i) f : N → N such that f(n) = n 2 (ii) f : N → N such that f(n) = n + 3 1 (iii) f : R → R such that f(x) = x 3 (iv) f : R → R such that f(x) = x 2 + 1 (v) f : N → N such tha…

(a) Let A = {a,b,c,d,e,f} and B = {1,2,3,4,5,6}. Determine whether each of following functions from A to B are invertible. If it is an invertible, find it’s inverse. (i) f1 = {(a,1), (b,2), (c,3), (d,4), (e,5), (f,6)} (ii) f2 = {(a,2)…

(b) Determine whether each of following functions are invertible. If it is an invertible, find it’s inverse. (i) f : R → R such that f(x) = 3x + 10 (ii) f : R → R such that f(x) = x 2 - 1

(iv) f : N → Y such that f(n) = 2n + 1, where Y = {y ∈ N : y = 4n + 3 for some n ∈ N}

(a) Consider f : R → R, f(x) = x 2 and g : R → R, g(x) = x + 5. (i) What is g ◦ f? (ii) What is f ◦ g

(b) Given f(x) = x 2 , g(x) = 2x and h(x) = x - 2 be functions from R to R. Find (i) f ◦ g (ii) f ◦ h (iii) h ◦ f (iv) (f ◦ g) ◦ h (v) f ◦ (g ◦ h)

Display the graph of following functions. (i) f(n) = n 2 + 4 from the set of integers to the set of integers

Display(ii) f(x) = x 3 + 1 from the set of real numbers to the set of real numbers the graph of following functions.

how many reflexive relation on A={1,2,3,4,5},could be definde

-151 mod 26 True Or false

injective functions always have inverse True or False

How many ways can be letters { 5.a. 4.b, 1.c} be arranged so that all letters of same kind are in a single block

Find the coefficient x to the power 4 * y cube & number of terms in the expansion of (3x – 10y)11

Determine whether the conclusion C is valid in the following premises without using truth table: H1: ¬Q, H2: P → Q, C: ¬ P

If f: A →B & g: B→C are two onto functions, then the mapping gof: A→C is also an onto function. Prove

In any Boolean algebra, Such that (a + b) (a’ + c) = ac +a’b = ac + a’b + bc

Determine the height of binary tree whose largest level order index is 2 to the power 5 + 33.

S.T if G is a polyhedral graph then there is a region of degree ≤ 5

show that if A and B are sets A u B = A n B a. by showing each side is a subset of the other set b. using a membership table

let A be the set of integers and C be the set of ordered pairs (x,y)£A×A such that y is not equal to zero define relation ~ on C(x,y)~(z,w) if yz=zw prove that defines an equivalence relation on C

How many transitive relatuon are there ona set with m elements If m=1 , m=2 and m=3 Solve the question

Prove that the relation of anti symmetric relation on a set with K elements are 2^k×3^(k^2_ k)÷2

Let Q(x) denote the statement "x + 1 = 2x". If the universe of discourse is all integers, what are these truth values? a) Q(0) b) Q(-1) c) Q(1) d) existential quantifier xQ(x) e) universal quantifier xQ(x) f) ~univers…

Find a number x < 77 such that x^37 = 24 mod 77. Show your work

Let P(x) be the statement “x likes subject mathematics,” where the domain for x consists of all students. Express each of these quantifications in English. a) ∃

Let H(x) be the statement “x is hardworking”, N(x) be the statement “x is naughty” and C(x) be the statement “ x is clever”, where the domain for x consists of all students. Use quantifications to express each of the following stateme…

Determine whether f is a function from Z to R if (a) f(n) = ±n (b) f(n) = √ n2 + 1 (c) f(n) = 1 n2−4

(i) Determine whether each of these functions from [a,b,c,d] to itself is one-to-one. (a) f(a) = b, f(b) = a, f(c) = c, f(d) =d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = d, f(b) = b, f(c) = c, f(d) = d (ii) Which functions…

(a) Give an explicit formula for a function from the set of integers to the set of positive integers that is (i) one-to-one, but not onto (ii) onto, but not one-to-one (iii) one-to-one and onto (iv) neither one-to-one nor onto

) Determine whether each of these functions is a bijection from R to R. (i) f(x) = -3x + 4 (ii) f(x) = 2x + 1 (iii) f(x) = x 2 + 1 (iv) f(x) = -3x 2 + 7 (v) f(x) = (x+1) (x+2)

) (i) Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3 and f(c) = 1. Is f invertible, and if it is, what is it’s inverse?

Let f : Z → Z be such that f(x) = x + 1. Is f invertible, and if it is, what is it’s inverse?

Find f ◦ g and g ◦ f, where f(x) = x 2 + 1 and g(x) = x + 2, are functions from R to R

(b) Find f ◦ g and g ◦ f, where f(x) = x 2 + 1 and g(x) = x + 2, are functions from R to R. (c) Find f + g and fg for the functions f and g given in part (b).

Draw the graph of the function f(n) = 1 - n 2 from the set of integers to set of integers.

Draw the graph of the function f(n) = 3n + 1 from the set of integers to the set of integers.

Draw the graph of the function f(x) = 3x 2 + 1 from the set of real numbers to the set of real numbers.

Determine whether each of these functions from [a,b,c,d] to itself is one-to-one. (a) f(a) = b, f(b) = a, f(c) = c, f(d) =d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c

Let P(x) be the statement “x likes subject mathematics,” where the domain for x consists of all students. Express each of these quantifications in English. a) ∃

5. Let C(x, y) be the statement “x is a friend of y,” where the domain for x and y consists of all people. Use quantifications to express each of the following statements. a) Everyone is a friend of everyone. b) Not everyone is a frie…

An explorer is captured by a group of cannibals. There are two types of cannibals: those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular canniba…

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 grou…

The following exercises relate to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people,A and B.Determine, if possible, what A and B ar…

The following exercises relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth,knaves who always lie, and spies who can either lie or tell the truth. You encounter three people,A…

A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that if the butler is telling the truth then so is the cook; the cook and the gardener cannot both be telling the tru…

For the given sequence 2, 6, 10, ... i- Write down the recursive formula for above sequence ii- Write down the closed formula for above sequence iii- Find the 11th term of the given sequence.

(1) Let Q(x, y) be the statement "x+y=xy." If the domain for both variables consists of all integers, what are the truth values? (a)Q(1,1) (b)Q(2,0) (c)∀yQ(1, y) (d)∃xQ(x,2) (e)∃x∃yQ(x, y) (f)∀x∃yQ(x, y) (g)∃y∀xQ(x, y) (h)∀y…

(2) Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. (a)∀x∃y(x^2=y) (b)∀x∃y(x=y^2) (c)∃x∀y(xy= 0) (d)∃x∃y(x+y=/=y+x) (e)∀x(x=/= 0 => ∃y(xy= 1)) (f)∃x∀y(y=/= 0 => xy= 1)…

(3) Express the negations of each of these statements so that all negation symbols immediately precede predicates. (a)∃z∀y∀xT(x, y, z) (b)∃x∃yP(x, y)^∀x∀yQ(x, y) (c)∃x∃y(Q(x, y)<=>Q(y,x)) (d)∀y∃x∃z(T(x, y, z)∨Q(x, y))

Show that (p∧q=⇒ r and (p=⇒r)∧(q=⇒ r) are not logically equivalent.

Suppose that a truth table in n propositional variables is specified. Show that a compound proposition with this truth table can be formed by taking the disjunction of conjunctions of the variables or their negations, with one conjunc…

There are 10 stalls for animals in an exhibition. Three animals; lion, pussycat and horse are to be exhibited. Animals of each kind are not less than 10 in number. What is the possible number of ways of arranging the exhibition?

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