quesition4 3. Consider the structure R =(IR, +, ×, <). We say that a set C CR is definable in R ifthere is a formula p(x) such that c E C if and only if p[c] is true in R. Show that the setC = {0,1} is definable in R.(Hint: Consider the polynomial whose solution is exactly 0 or 1.)4. Consider again the structure R(IR, +, ×, <) and let P(x, y) be the predicate x < y.Express the predicate using a formula p(x, y) involving only + and × and not involving <.That is, find a formula p(x, y) such that P(x, y) and p(x, y) are logically equivalent in Rwith p(x, y) not involving the symbol <.

The solution for the above given question is given below:

4. Consider again the structure R = (IR, +, x, <) and let P(x, y) be the predicate a < y. Express the predicate using a formula p(x, y) involving only + and x and not involving <. That is, find a formula p(x, y) such that P(x, y) and p(x, y) are logically equivalent in R with p(x, y) not involving the symbol <.

4. Consider again the structure R = (IR, +, x, <) and let P(x, y) be the predicate a < y. Express the predicate using a formula p(x, y) involving only + and x and not involving <. That is, find a formula p(x, y) such that P(x, y) and p(x, y) are logically equivalent in R with p(x, y) not involving the symbol <.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

Assume a generic recursive relation is defined as : A[i] = p*A[i-1] + q where p and q are coefficients of the relation. Write a function that takes the first three numbers in this relation as a list, such as [A[0], A[1], A[2]] and computes p and q, and returns A[3]. For example, given [1,3,5] your function should return 7, because the numbers 1,3,5 are produced by the series: A[i] = A[i-1] + 2 where p is 1 and q is 2, and then the next number in the series becomes 7. You can assume that p and q are always integers. You can also assume that p and q are determinable in any test case. """ def recursionSolver(values): return # Remove this line to answer this question.
Assume a generic recursive relation is defined as :A[i] = p*A[i-1] + qwhere p and q are coefficients of the relation. Write a function that takes the first three numbers in this relation as a list,such as [A[0], A[1], A[2]]and computes p and q, and returns A[3]. For example, given [1,3,5] your function should return 7, because the numbers 1,3,5 are produced by the series:A[i] = A[i-1] + 2where p is 1 and q is 2, and then the next number in the series becomes 7. You can assume that p and q are always integers. You can also assume that p and q are determinable in any test case.""" def recursionSolver(values):return # Remove this line to answer this question. İn python.
Q # 3; (a). The relation ‘R’ is defined on the set M=(4,5,6,7) as the following;R={(5,5),(5,6),(5,7),(6,5),(6,6),(6,7)} Check is it reflexive , symmetric and transitive. (b). Let the string having length eight of an arrays is defined as ; 01001010 , 01101101Apply the definition of Tautology on it. Q # 4; (a). Let ‘T’ and ‘S‘ are two non-empty sets which are defined as; T={x ; -1 ≤x≤1}=S and modulus function f: T→S defined by; f(x)=x|x| Check it for injective and also draw its graph(b). Simplify by using Boolean algebra definition; AB+BC+B ̅C=AB+C Q # 5. Given a simple graph G as below in figure ; Check it for the following ;Find its chromatic number which must be threeFind the degree of the graph and write it in degree sequence Label its edges and also write down its in vertex form
Let R ⊆ A × A be a a binary relationship defined in a non-empty set A. Let B be any set so that B⊆ A. Define the set Rb as RB := R ∩ (B × B). 1.) How to show that if R is reflexive, symmetric and transitive then Rb also is. 2.) Is the opposite true? Demonstrate truth or falseness. 3.) If R is antisimetric, is Rb antisimetric too?
Let A ={1, 4, 7, 10, 6, 5}, B = {4, 8, 10, 2, 9}, and C = {2, 3, 4, 5} Write the Venn diagram representation of the following sets. What is the set cardinality of the following set operations: B - C
Write your own example of a set P containing 5 ordered pairs (x, y) such that P is not a function.
Is each of the following statements true or false? Please explain. a. If a signature σ is infinite (i.e. contains infinitely many symbols), then a σ-formula can have infinite length. b. Let σ := (c, f, ·, <) be a signature, where c is a contant symbol, f is a unary operation symbol, · is a binary operation symbol, and < is a binary relation symbol. Then ϕ(y, z) := ∀x(f(x) · c < x) is an extended σ-formula whose interpretation in each σ-structure is a binary relation. c. For any signature σ and any σ-structure A := (A, σA), if a set D ⊆ Ak is definable (A-definable here), then it is P-definable for some finite P ⊆ A.
Given: Relation R = {(a,a), (a.b), (b,a). (b,b). (c,c)} and Set S = {a, b, c} Complete Relation R to be Transitive with Set S
For function f from {a, b, c, d} to {1, 2, 3, 4, 5}. f(a) = 4, f(b) = 5, f(c) = 1 and f(d) = 3, The set containing the domain has the following elements
Let A be a non-empty funute set and let S = P(A). Define relation R ⊆ S × S by: xRy ⇔ |x| = |y|, where | · | denotes the set cardinality.(i) Prove that R is an equivalence relation. (ii) Find all equivalence classes for the case A = {1, 2, 3}.
Answer itthe nfa. L1 = {u ∈ Σ∗| u ends with aa}.L2 = {u ∈ Σ∗| u ends and begins with different letters }.L3 = {u ∈ Σ∗| u contains abba}.L4 = {u ∈ Σ∗| u is of the form anbamfor n,m > 0}. Given the above languages:(a) Use the set operators ‘union’ and ‘complement’ to describe L5 = L1 ∩ L2.(b) Prove that L5 is regular.(c) Construct an NFA M that accepts L5, and prove its correctness.?
Suppose (X[mX]+Y [mY ]) > 0. What can you infer about each of the four subproblems (namely, (X1, Y1), (X1, Y2), (X2, Y1), and (X2, Y2))? Clearly identify the ones that do not have to solved anymore, and those that still have to be solved. Briefly justify.