Solved: (a) Let P(x) be the statement x 2 ≥ x. (i) What are truth values of the propositions P(1) , P(-1) , P(0) and P( 1 2 )? (ii) What is the truth value of the proposition ∀xP(x), where the domain consists of all real numbers? (iii) What is the truth value of the proposition ∀xP(x), where the domain consists of all integers? (b) Let Q(x,y,z) be the statement x 2 + y 2 = z 2 . What are the truth values of (i) ∀(x,y,z) Q(x,y,z) (ii) ∃(x,y,z) Q(x,y,z) Where the domain consists of all real numbers.

The following table shows the income distribution of 600 families. Find the minimum income of the riches 30% families. Also the limits of income of middle 50% of families, to the nearest rupees. Income Below 75 75- 1…

a) Suppose P (x, y) denotes the equation8, what will the truth values of the Propositions P (2,2), P (0,4).

Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

Let 8Z be the set of all integers that are multiples of 8. Prove that 8Z has the same cardinality as 3Z, the set of all integers multiples of 3.

We assume the question is: What is mathematical foundation for computer science? Solution: The mathematical side concentrates on areas where computers are used, or which are relevant to computer science, namely algebra, general topo…

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Prove that the relation R:{(x,y)|x-y is divisible by 3} is an Equivalence relation.(R is defined over Z) L2 CO2 10

(a) By the definition of piece arrow- P\downarrow Q=P↓Q= ~(P\lor Q)(P∨Q) P\downarrow QP↓Q =~(P\lor P)(P∨P) We have derived that P\downarrow PP↓P is logically equivalent with ~P ~P=P&b…

Show that the following logical equivalences hold for the Peirce arrow ↓, where P ↓ Q ≡ ∼(P ∨ Q). a. ∼P ≡ P ↓ P b. P ∨ Q ≡ (P ↓ Q) ↓ (P ↓ Q) c. P ∧ Q ≡ (P ↓ P) ↓ (Q ↓ Q) H d. Write P → Q using Peirce arrows on…

F. Show using the rules of resolution/inference, that no single assignment of truth values to p, q, r makes all the disjunctions p V-9, p V-9,9 V r, qVT, V revaluate to true. Proof using truth na

F. How many different sequences, each of length r, can be formed using elements from A if (a) elements in the sequence may be repeated? (b) all elements in the sequence must be distinct?

Given the following 2 premises, 1. 𝑝 → (𝑞 ∨ 𝑟) 2. 𝑞 → 𝑠 Prove 𝑝 → (𝑟 ∨ 𝑠) is valid using the Proof by Contradiction method.

Use the Principle of Mathematical Induction to prove that (2𝑛)! /(2n)𝑛! is odd for all positive integers.

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Use the truth tables method to determine whether (¬p ∨ q) ∧ (q → ¬r ∧ ¬p) ∧ (p ∧ r) is satisfiable.

Suppose that A,B and C are sets such that A is the improper subset of B and B is the improper subset of C

(p^q)v(P ^ Q ^ R)

Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

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 antisy…

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Use the Principle of Mathematical Induction to prove that ((2n) !)/(2^n n !)((2n)!)/(2 n n!) is odd for all positive integers.

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Given the following 2 premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) is valid using the Proof by Contradiction method.

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Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

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Given The Following 2 Premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) Is Valid Using The Proof By Contradiction Method. [Hint: Use A Combination Of Equivalence Laws And Rules Of Inference To Solve This Question]

Construct the network who's activity and the relationship Activity:A,D,E can start simultaneously Activity:B,C>A;G,F>D,C;H>E,F

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Construct the network who's activity and the relationship Activity:A,C,D can start simultaneously Activity:E>B,C;F,G>D;H,I>E,F;J>I,G;K>H;B>A

Construct the network who's activity and the relationship Activity:A

Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive

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which of the following statements are true? justify your answers i) the contrapositive of "not a ⇒ not b' is "a&b', where a and b are two statements. ii) any set can be represented by the listing method

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 …

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{F} How many different sequences, each of length r, can be formed using elements from A if (a) elements in the sequence may be repeated? (b) all elements in the sequence must be distinct?

{F} Show using the rules of resolution/inference, that no single assignment of truth values to p, q, r makes all the disjunctions p V-9, p V-9,9 V r, qVT, V revaluate to true. Proof using truth na

{F} Show that the following logical equivalences hold for the Peirce arrow ↓, where P ↓ Q ≡ ∼(P ∨ Q). a. ∼P ≡ P ↓ P b. P ∨ Q ≡ (P ↓ Q) ↓ (P ↓ Q) c. P ∧ Q ≡ (P ↓ P) ↓ (Q ↓ Q) H d. Write P → Q using Peirce arrow…

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{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…

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Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B ∪ C by showing each side is a subset of the other side. using a membership table.

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Venn diagram (A ∩ B) ∪ (A ∩ C) (A ∩ ) ∪ (A ∩ )

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