Solved: 1. Expand in a two-element universe (the elements are named 'a' and 'b') (a) ~(x) ((Fx v Gy) v Ka) (b) (x) ~ (Kx v Ka) (c) (Ex) (Cy v (Fx --> ~Ga)) 2. For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Make sure you draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line. You can write the wffs with the vertical lines on a piece of paper and submit a photo of it. (a) (x) (y) (z) ((Fxy --> Gxy) v (Hxa --> Hzbu)) (b) (Ex) (Ey) Hxxy --> (z) (Gz v Hx) (c) (z) (Ex) (y) (Axayzw v Bxycuvz)

{F} Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor an…

{F} 2. Find the transitive closures of these relations on {1, 2, 3, 4}. a) {(1, 2), (2,1), (2,3), (3,4), (4,1)} b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)} c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)} d) {(1, 1), (1,4), (2…

{F} Construct the Combinatorial Circuit of the given output. (p∨q) 2. (p∧ ~q) (p→~q)→(p∨q)

USE THE METHOD OF DIRECT TO PROVE: If a is an odd integer, then a’2 + 3a +5 is odd.

This question has 2 parts. Part 1: Suppose that F and X are events from a common sample space with P(F) 6= 0 and P(X) 6= 0. (a) Prove that P(X) = P(X|F)P(F) + P(X|F¯)P(F¯). Hint: Explain why P(X|F)P(F) = P(X ∩ F) is another way of wri…

Let p and q be the following propositions: p: Angela misses the final exam. q : Angela passes the course. Write the following propositions using p, q and negations and logical connectives: (a) Angela either m…

250 members of a certain society have voted to elect a new chairman. Each member may vote for either one or two candidates. The candidate elected is the one who polls most votes. Three candidates x, y z stood for election and when t…

whether it is the linearly ordered set or not. Let A = $1, 2, 3, 4, 6, 9} and let R be the relation on A defined by "x divides y " written x/y. a) Write R as a set of ordered pairs b) Drawits directed graph. c) Find ind…

Solution: a) p∧q b)p∧¬q c)¬p∧¬ q d)p∨q e)p→ q f)(p∨q)∧(p→ ¬ q) g)p↔q

Let p and q be the propositions p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical con- nectives (including negations). a) It is below freezing and snowing. b)…

Determine whether each of these compound propositions is satisfiable. a) (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q) b) (p → q) ∧ (p → ¬q) ∧ (¬p → q) ∧ (¬p → ¬q) c) (p ↔ q) ∧ (¬p ↔ q)

Explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r is true and at least one is false, but is false when all three variables have the same truth value. Expert's ans…

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r…

Let p and q be the propositions “The total amount is discounted” and “The items have been packed,” respectively. Express each of these compound propositions as an English sentence.

What is p ⊕ p ? true or false

a) i) Give an inductive formula for the sum of the first n odd numbers: 1 + 3 + 5 + ... + 2n -1 Show your induction process. ii) Use the proof by mathematical induction to prove the correctness of your inductive formula in i) a…

Consider the following premises: 1. A \to→ (B \to→ A) is a Theorem of Propositional Calculus/Logic (i.e. it’s logically valid), for all statement forms A and B. Suppose then that the following are the temporary axioms…

. 1. Let D = {3, 9, 15}, E = {3, 15, 9}, F = {3, 3, 9, 15, 15, 15}. What are the elements of D, E, and F? How are D, E, and F related? 2. How many elements are in the set {1, {2}, [1, 2}}? 3. For each positive int…

1. Write the following sentences in the form of if... then and then find the inverse, converse, and contrapositive [8] i) You can become successful and happy unless you lead a lazy life. ii) We enter the password or we click on the…

a) The aim is to determine whether the set ∅ is a power set of a set or not. By definition, given a set S, the power set of S is a set of all subsets of the set S. The power set of S is denoted by P(S). The power set of every set in…

Determine whether each of these sets is the power set of a set. where a and b are distinct elements. a) ∅ b) {∅.{∅}} c) {∅, {a}. {∅, a)} d) {∅, {a}. {b}. {a, b}}

Let P(x) be the statement “x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in English. a) ∃xP(x) b) ∀xP(x) c) ∃x ¬P(x) d) ∀x ¬P(x)

Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let…

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is…

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the…

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not th…

Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.

Show that ∃x(P(x) ∨ Q(x)) and ∃xP(x) ∨ ∃xQ(x) are logically equivalent.

Show that ∃xP(x) ∧ ∃xQ(x) and ∃x(P(x) ∧ Q(x)) are not logically equivalent.

Consider the statement form (P \downarrow Q) \downarrow R Now, find a restricted statement form logically equivalent to it, in a) Disjunctive normal form (DNF). b) Conjunctive normal form (CNF).

Consider the statement form (P↓Q)↓R. Now, find a restricted statement form logically equivalent to it, in a) Disjunctive normal form (DNF). b) Conjunctive normal form (CNF).

A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W, X, Y and Z. We define the relation: A Sesotho word x is related to another Sesotho word y if x begins with the same letter as y. Determine whet…

Let O be the set of odd numbers and O’ = {1, 5, 9, 13, 17, ...} be its subset. Define the bijections, f and g as: f : O \to→ O’, f(d) = 2d - 1, \forall∀ d \in∈ O. g : \NuN \to→ O, g(n) = 2n + 1, \foral…

Consider the following premises: 1.A-->(B-->A) is a theorem of proportional calculus for all statements A and B. 2.suppose then that the following are the temporary axioms. a)w b)y c)y-->z Using the log…

If A={1,3,5,7} and B={1,3,7}. Is set B a proper subset of set A? Explain. Expe

Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this proof and discuss/explain each step.

If A={1,2,3}, identify the power set of the given set A. P(A)= P(A)={ } and P(A)=

show that (A ∪ B)\C ⊆ A ∪ (B\C)

Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this proof and discuss/explain each step.a) i) b_0=1, b_1=2, b_n=2^nb 0 =1,b 1 =2,b n =2 n Let a_1=b_0=1, a_2=b_1=2, ..., a_{n+1}…

Show that the argument form with premises (p ∧ t) → (r ∨ s), q → (u ∧ t), u → p, and ¬s and conclusion q → r is valid by first using exercise 11 and then using rules of inference from table 1.

If you are at least 40 years old, a natural-born citizen of the Philippines, and you are a registered voter, then you are eligible to be elected as the Philippine President. a: You are at least 40 years old. b: You are a natural-…

Show that the explicit sequence {yn}∞n=0 such that yn = A(2n )+ B(-1)n for any nonzero constants A and B is the solution of the recurrence relation yn = yn-1 + 2yn-2 for n >1.

Let T(x) = x is a tool, C(x) = x is in the correct place and E(x) = x is in excellent condition. Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. i) Everything is in…

3. A box contains 4 bad and 6 good tubes. Two are drawn out from the box at a time. One is tested and found to be good. What is the probability that the other one is also good?

An unbiased coin is tossed twice. If A is the event: both head or tail have occurred and B is the event: at most one tail is observed find : i) P(A) ii) P(B) iii) P(A\B) iv) P(B\A)

5. What is the probability that exactly eight 0 bits are generated when 10 bits are generated with the probability that a 0 bit generated is 0.9, the probability that a 1 bit generated is 0.1, and the bits are generated independently?

List the members of these sets. (a) {x | x is a real number such that x 2 = 5} (b) {x | x is a positive integer less than 15} (c) {x | x is the square of an integer and x < 200} (d) {x | x is an integer such that x 2 = 5}

It is impossible for a valid argument to have true premises and

draw the hasse diagram for the set S=[0,10,20,30,40]

How many edges are there in a graph with 15 vertices each with degree 8?

6. Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consists of the students in your class and second, let it consists of all p…

Let f : G → G0 be a group epimorphism, and let H be the normal subgroup that be the Kernal of the epimorphism. Then, prove that G0 is isomorphic to G/H.

A pair of dice is loaded. The probability that a 4 appears on the first die is 2/7, and the probability that a 3 appears on the second die is 2/7. Other outcomes for each die appear with probability 1/7. What is the probability …

Let the proposition p be T and proposition q be F. Find the truth value of the following a. p^›q

((𝑝 → 𝑞) ∧ (𝑞 → 𝑟)) → (𝑝 → 𝑟)

Prove by induction (make sure to show the base case, the inductive hypothesis, and all steps in the proof) Sum (from k=1 to N) of k is less than N^2 for all N >= 2

(a) Given an array A of integers, sorted into ascending order, design an efficient algorithm for finding is there is any array subscript i where A[i] == i. What is the worst case run time of your algorithm (in 'Big-Oh' notation)?…

Prove that the fourth power of an odd integer is expressible in the form 16n + 1 for n ∈ Z.

Define an by a0 = 1, a1 = 2, a2 = 4 and an+2 = an+1 + an + an−1, for n ≥ 1. Show that an ≤ 2n for all n ∈ N.

If Fn is the nth Fibonacci number, prove that Fn+1Fn−1 − F2n = (−1)n.

Let a and n be positive integers with a > 1. Prove that, if a^n + 1 is prime, then a is even and n is a power of 2.

Show that if all three of p, p + 2 and p + 4 are prime, then p = 3.

Use the Euclidean algorithm to compute (2059, 2581) and to express this quantity as a linear combination of 2059 and 2581.

Show that every nonzero integer can be uniquely expressed as ak3^k + ak−13^(k−1) + · · · + a13 + a0 where ai ∈ {−1, 0, 1} and ak ≠ 0.

Prove that the product of any two integers of the form 6k + 1 is of that same form.

Prove that any prime p > 3 is either of the form 6k + 1 or of the form 6k + 5 for some integer k.

Prove that there are infinitely many primes of the form 6k + 5. That is, consider the primes which has a remainder 5 when divided by 6. Prove that there are infinitely many such primes.

Prove that if k and ℓ are posetive integers then k^2 − ℓ^2 can never be equal to 2.

If a is an odd integer prove that a^2 − 1 is always divisible by 8.

Show that for any positive number a and b, (a + b)/2 ≥ √ab

Let x be an integer. Prove that if x^2 − 6x + 5 is even then x must be odd.

Show that a^2 + a^4 ≡ 0(mod 5) if a ≡ 2(mod 5) or a ≡ 3(mod 5) or if 5 divides a.

Show that if a, b, c, and d are integers such that a|c and b|d, then ab|cd.

Prove that if n is an odd positive integer, then n^2 ≡ 1 (mod 8).

Prove the logical equivalence of the following statements, ~(p V ~q) V (~p ^ ~q) ≡ ~p

Determine which of the following pair of statement are logically equivalent: (r V p) ^ ( ( ~r V (p^q) ) ^ (r V q) ) and p ^ q

Write truth table for the statement forms: ~(p ^ q) V (p V q)

Construct the truth table for the statements: (pVq) V (~p^q) → q

Write each of the following three statements in the symbolic form and determine which pairs are logically equivalent: a) If it walks like a duck and it talks like a duck, then it is a duck. b) Either it does not walk like a duck or it…

Rewrite the statement in the if-then form: A necessary condition for this computer program to be correct is that it does not produce error messages during translation:

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.

Use truth tables to show that the following forms of argument are invalid:p → q, ~p, hence, ~q (inverse error)

Use truth tables to determine if the argument forms are valid: p → r q → r hence, p V q → r

Use inference rules to deduce the following conclusions from the following sets of premises: a) Premises: p ∨ q q → r, p ∧ s → t, ¬r, ¬q → u ∧ s Conclusion: t b) Premises: (p ∧ t) → (r ∨ s), q → (u ∧ t), u → p, ¬s Conclusion:…

Construct a truth table for each of these compound propositions: i) (p ⇔ q) ⇔ (r ⇔ s) ii) (p ⨁ q) ∨ (p ⨁ ¬q)

Compute the value of the double summation: ∑i ^ 4 = 1 ∑j=i ^4 (3i + j)

Prove that if x^3 is irrational, then x is irrational

Show that these compound propositions are tautologies: (1) (¬q ∧ (p → q)) → ¬p (2) ((p ∨ q) ∧ ¬p) → q

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Question ID: mtid-5-stid-8-sqid-4108-qpid-2807