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- 5. Which relation on the set {1, 2, 3, 4} is an equivalence relation and contain {(1, 2), (2, 3), (2, 4), (3, 1)}. 6. Find the transitive closures of the relation {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)} on the s…
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- Q.1 Prove by contrapositive that if n = a*b, where a and b are positive integers, then a ≤ √n or b ≤ √n
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- {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…
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- 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…
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- Solution: a) p∧q b)p∧¬q c)¬p∧¬ q d)p∨q e)p→ q f)(p∨q)∧(p→ ¬ q) g)p↔q
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- 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…
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- 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…
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- . 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…
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- 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…
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- 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)

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