Problem 1the sequence of fibonacci numbers is defined by

Question

PROBLEM 1.

The sequence of Fibonacci numbers is defined by

f₀ = 0, f₁ = 1, and f_n = f_n-1 + f_n-2, for n > 1.

The sequence of Lucas numbers is defined by

l₀ = 2, l₁ = 1, and l_n = l_n-1 + l_n-2, for n > 1.

Prove that

fn + fn+2 = ln+1,

whenever n is a positive integer, where f_i and l_i are the ith Fibonacci number and ith Lucas number, respectively.

PROBLEM 2.

For each of the following relations on the set Z of integers, determine if it is reflexive, symmetric, anti-symmetric, or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

R = {(a, b) | a² = b²}.

S = {(a, b) | | a - b | ≤ 1}.

1

PROBLEM 3.

Prove that {(x, y) | x - y ∈ Q} is an equivalence relation on the set of real numbers, where Q denotes the set of rational numbers.

Give [1], [1/2], and [π].

PROBLEM 4.

Prove or disprove the following statements:

Let R be a relation on the set Z of integers such that xRy if and only if xy ≥ 1. Then, R is irreflexive.

Let R be a relation on the set Z of integers such that xRy if and only if x = y + 1 or x = y - 1. Then, R is irreflexive.

Let R and S be reflexive relations on a set A. Then, R - S is irreflexive.

PROBLEM 5.

Let R be the relation on Z⁺ defined by xRy if and only if x < y. Then, in the Set Builder Notation, R = {(x, y) | y - x > 0}.

Use the Set Builder Notation to express the transitive closure of R.

Use the Set Builder Notation to express the composite relation Rⁿ, where n is a positive integer.

PROBLEM 6.

Give an example to show that when the symmetric closure of the reflexive closure of the transitive closure of a relation is formed, the result is not necessarily an equivalence relation.