Solved: Use symbols to write the logical form of each argument. If the rule is valid then identify the rule of inference that guarantees its validity, otherwise state whether converse or inverse error is made.

Question five For each of the following sentences, write the sentence in logical notation , negate the sentence , and say whether the sentence or its negation is true. a) Given any integer, there is a larger integer.

Question one Prove by induction that the following statements are true for all integers n a) 12x2+22x3+…..+n2(n+1)= n(n+1)(n+2)(3n+1)/12 b)4007n-1 is divisible by 2003

X be a non-empty set and let R be an equivalence relation on X. For each x ∈ X, define [x]={y∈X suchthatxRy} to be the equivalence class of x. Here x R y means (x, y) ∈ R. SupposethatA=[x]andB=[y]. ProvethatifA∩B̸=∅,thenA=B.

Find domain and range of (answers should be subsets of R): f(x)= 3/2x -1

Find domain and range of (answers should be subsets of R): f(x)= 1/(5x-6)

Find domain and range of (answers should be subsets of R): f(x)= x/(x^2+1)

Find domain and range of (answers should be subsets of R): f(x)= 3/(2x -1)

3. Prove or give a counterexample to the following: For a set A and binary relation R on A, if R is reflexive and symmetric, then R must be transitive as well.

Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements. Prove that ∼ is an equivalence relation. Write down all equivalence classes of ∼.

Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}. Now, define f : X → C by the rule f(x) = [x] for all x ∈ X. Prove that…

Let X be a non-empty set, and let R be an equivalence relation on X. Let C be the set of all equivalence classes of R. So C={A⊆X such that A=[x] for some x ∈ X}. Now, define f : X → C by the rule f(x) = [x] for all x ∈ X. Suppose X …

Let X = {1,2,3}. Define a relation ∼ on P(X) by A ∼ B if A and B have the same number of elements. Prove that ∼ is an equivalence relation and write down all equivalence classes of ∼.

How many ways can you assign 4 caretakers each of which look after 2 lighthouses for a year (for a total of 8 lighthouses), and then assign each two lighthouses in the next year so that none of them tend the same two lighthouses both …

Find a generating function in closed form for the sequence: {1,2,3,4,1,2,3,4,1,2,3,4,...}

Prove that for all sets A and B: A⊆B ⇐⇒ A∪B=B.

Prove that A − (B ∪ C) = (A − B) ∩ (A − C), for sets A, B and C.

Prove that for all sets A and B, A ∪ B^c = B^c ∪ (A ∩ B).

Prove that A − (B ∪ C) = (A − B) ∩ (A − C),

Prove that 3n2 − n + 10 is an even integer for all integers n. (Hint: Prove the statement true when n is odd, then prove it is true when n is even.)

Define the relation R as R = {(a, b)|a, b ∈ Z, 4 divides a − b}. Show that R is reflexive, transitive and symmetric.

Let X = {0, 1, 2}. Consider the power set P(X). Recall that P(X) = {A such that A ⊆ X}. Determine if each statement is true or false. (a) X ∈ P(X). (b) ∅∈P(X) (c) X ⊆ P(X). (d) {0,2}∈P(X) (e) {0,2,1} ∈ X (f) {0}⊆X (g) {{0}} ⊆ P(X)

Prove that for all positive integers n 3^n +7^n −2 is divisible by 8

Prove that for all integers n with 1≤n≤10,n^2 −n+11 is prime.

Describe two situations that can be modelled with a Graph. Describe the graph model that you will use

Describe two situations that can be modelled with a Tree. Describe the graph model that you will use.

Describe two situations that can be modelled with a Graph. Describe the graph model that you will use. Describe two situations that can be modelled with a Tree. Describe the tree model that you will use.

Let R = {(2, 2), (2, 4), (3, 3), (3, 6), (3, 12), (4,2), (6, 3)} where if a is a factor b} (i) Draw the digraph for the relation P. (ii) Find the domain and range of the relation.

Q:1. Determine whether each set is a function from X = {1,2,3,4} to Y = {a,b,c,d}. If it is a function, find its domain and range, draw its arrow diagram, and determine if it is one-to-one, onto or both. a) {(1,a,),(2,a),(3,c),(4,b)} …

Q : 1) Determine whether each function is one-to-one. The domain of each function is the set of all real numbers. If the function is not one-to-one, prove it. Also, determine whether f is onto the set of all real numbers. If f is not …

Q : 5). If f is function from A to B and g is function B to C and both f and g are onto. Show that go f is also onto. Is go f one-to-one if both f and g are one-to-one. Q : 6) Let f, g and h: R → R be defined by (R is the set of rea…

Construct the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-0011, two ca…

Discrete Mathematics - Languages and Machines Need to solve Finite State Automaton i.e, Your task is to design a binary finite state automaton (FSA) to accept all strings that represent valid messages (for your particular codes and pa…

For any two sets A and B , in a universal set U , prove that A ⊆ B ⇔ A ∪ B = B.

a) Simplify the Boolean function F = AB + (AC)′ + AB ′C(AB + C).

Implement the 3-variable function F (A,B,C) = (0,2,4,7) with a multiplexer

a. Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).

(A \ B )∪ C = A \ (B∩ C )for any three sets A,B and C Is the statement true or false? Give justification in support of your answer.

GreenRecycle.com is a recycle firm whose main business is recycling paper. There is a particular standard to grade the recycled papers in the firm, which grades the papers as P11, P12, P13, P21, P22, P31 and P32. a) Model the set of …

Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).

Let A = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} and B = {1; 2; 3; 4}. Let R be the relation on P (A) de fined by: For any X,Y element in P(A), XRY if and only if X-B = Y-B. How many equivalence classes are there? Explain

(p->(q^r))^(not p->not q^not r))

Draw a Venn diagram of sets B,A and C where A . ⊆ ,B A ∩ C ≠ φ, B∩ C = φ What is the universal set you have chosen? Justify your choice of sets in the diagram.

Define (a) Graph; (b) Null graph; (c) Isolated vertex; (d) Pendant vertex; (e) Pseudo-graph; (f) Directed graph; (g) Adjacent nodes; (h) Incident edges;

Let A be an 8*8 boolean matrix (i.e.,every entry is 0 or 1).if the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further show that…

Which of the following statements are true and which are false? Give reasons for your answer. i) (∼ p∨q) and (p ← q) are logically equivalent. ii) (m+n)! > m!+n! for m, n any positive integers. iii) The number of distributions of m in…

a) Which of the following sentences are statements? Give reasons for your answer. i) Are the buses running today? ii) What a pleasant weather! iii) x2 +1 = 0 for a real value of x. iv) Every odd number is a prime. b) Let t(n) denote t…

a) Let Fn denote the nth Fibonacci number. Show by induction that every natural number is expressible as a sum Fn1 +Fn2 +···+Fnk where ni −ni+1 > 1 for each i ≥ 1. b) Find the number of ways of tying up 7 different books into 4 bundle…

a) Make a table of the values of the Boolean function f(x1,x2,x3) = x2 ⊕(x1 ∧x3) Write the function in DNF using the table. b) Find the general form of the solution to a linear homogeneous recurrence with constant coefficients for whi…

a) Let A be an 8×8 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further, show…

A) Find the generating function of the recurrence an = 4an−1 −4an−2 +1 with initial conditions a0 = 1, a1 = 1. B) Express 3x4 +2x3 −2x2 +x in terms of [x]4, [x]3, [x]2 and [x]

Find the number of equivalence relations that can be defined on a set of 6 elements.

Explain 1. Game Tree 2. Decision Tree

Mathematical Induction. Suppose that you know that a cyclist rides the first kilometre in an infinitely long road, and that if this cyclist rides one kilometre, then she continues and rides the next kilometre. Prove that this cyclist …

Determine if the statements are valid arguments. I just got my ticket to the concert but if the number of tickets sold is more than 75% of its capacity, I will not want to go because it will be too crowded and I don’t like it. If my …

determine that wheter the functions from real numbers to real numbers are one to one f(n)=n^3 f(n)=n^2+1

In a lottery, players win a large prize when they pick four digits that match, in the correct order, four digits selected by a random mechanical process. A smaller prize is won if only three digits are matched. What is the probability…

If we reduce the number of elements by two, the number of permutations reduces thirty times. Find the number of elements.

How many 8-bits sequences that start with the same two bits or their fourth and fifth bits are equal or end with the same two bits are there?

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain

Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r

Determine if the following argument is valid using truth tables. p −→ r r _______ ∴ p

Determine if the following argument is valid using truth tables p ←→ q p∧ ∼ q _________ ∴ r

Determine if the following argument is valid using truth tables p → q p → r __________ ∴ p → (q ∧ r)

Determine if the following argument is valid using truth tables. p → (q ∨ r) ∼ q __________ ∴ p → r

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain

write the following boolean expressions in an equivalent sum of product canonical form in three variables x1, x2, and x3: 1. x1*x2 ? 3. (x1+X2)'*X3

According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calc…

In how many ways can one select 7 member committee from 10 distinct persons if only three persons qualify to be chairperson?

Prove that the conditional proposition and its contrapositive are logically equivalent suing the truth table.

Find a counter-example to the following statement: For all real numbers x > 1, 1/x^2+1 ≤ 1/2^x+1

Prove that for all integers a, b, c if a|b then ac|bc

Prove that for all integers a, b, c such that c =/= 0, if ac|bc then a|b.

Prove that for all integers n, n(n + 2)(n + 4) is divisible by 3.

Draw the graph represented by the adjacency matrix . (0 0 1 0 1 0 1 0 0)

In how many ways can one select 7 member committee from 10 distinct persons if only three persons qualify to be chairperson?

Prove by mathematical induction the formula (1^3+2^3+3^3+4^3+....n^3)=(n^2(n+1)^2)/4

Prove that the conditional proposition p->q and its contrapositive ~q ->~p are logically equivalent using the truth table.

Given that p and q are propositions construct the truth table of p -> q p <-> q

Prove that the conditional proposition p⟶q and its contrapositive ~q⟶~p

Given that p and q are propositions construct the truth table of p⟶q p<->q

Recall that a real number x is rational if x = p/q for integers p, q with q = ̸= 0. Prove that if x is rational then 1/(2x+1) is rational. Then prove that if 1/(2x+1) is rational then x is rational.

According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calc…

6.which of the following statement is true a.(p∨q)∧(p∨r)=p∨(q∧r) b.(p∧q)∧(p∨r)=p∨(q∧r) c.(p∧q)∨(p∨r)=p∨(q∧r) d.∼(p∨q)=∼(p∧∼q) 7. ____ reads â€œthe goods are standard if and only if the goods are expensiveâ€ a.p↔q b.∼p∧q c.∼∼q…

b) Given that p and q are propositions construct the truth table of; (4 Marks) i. p->q ii. p<->q

(p∧q)=(q∧p) and (p∨q)=(q∨p) implies an.............. Answers Options a. Associative laws b. Distributive Laws c. Commutative Laws d. Idempotent Laws

Draw the graph represented by the adjacency matrix . (001 010 100)

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(3, 2),(2, 1)} Is R transitive? Symmetric? Reflexive?

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(1, 3)} Is R transitive? Symmetric? Reflexive?

Let C = {1, 4, 5} and D = {2, 7}. • List the elements of C × D. • If there are 140 elements in D × A, how many elements are in A? • Give an example of a relation from D to C that is not a function. Draw a graph of the relation. Expla…

Prove by mathematical induction that: Where "E" is the summation icon. n i E E j = 1/6n(n+1)(n+2) i = 1 j=1

Use symbols to write the logical form of each argument. If the rule is valid then identify the rule of inference that guarantees its validity, otherwise state whether converse or inverse error is made.

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Question ID: mtid-5-stid-8-sqid-15-qpid-8