Suppose that

v

$$

$=$

$$

v

$1$

$,$

v

$2$

$,$

$.$

$.$

$.$

$,$

v

n

$$

v

$=$v

$1$

$$

$,$v

$2$

$$

$,...,$v

n

$$

$$ and

u

$$

$=$

$$

u

$1$

$,$

u

$2$

$,$

$.$

$.$

$.$

$,$

u

n

$$

u

$=$u

$1$

$$

$,$u

$2$

$$

$,...,$u

n

$$

$$ are a pair of

n

n$-$dimensional vectors. Assume that each component of the vector is a real number, so

v

$$

v

and

u

$$

u

are both members of the set

R

n

R

n

$($set of all n$-$tuples of real numbers$).$

We will say that

v

$$

v

and

u

$$

u

are "almost the same" when every component of

v

$$

v

is close to every component of

u

$$

u

$.$ That is$,$

v

$1$

v

$1$

$$

is close to

u

$1$

u

$1$

$$

$,$

v

$2$

v

$2$

$$

is close to

u

$2$

u

$2$

$$

$,$ etc $($practically speaking, "close" means that their absolute difference is small$).$

Assume that we are given the predefined predicate

CloseTo

$($

x

$,$

y

$)$

CloseTo$($x$,$y$)$ and the integer constant

n

n$.$ Use them to write a formal definition of the new predicate

AlmostTheSame

$($

v

$$

$,$

u

$$

$)$

AlmostTheSame$($

v

$,$

u

$)$ which asserts that

n

n dimensional vector

v

$$

v

is almost the same as

u

$$

u

$.$

Tip: It is not legal to say

i

in

v

$$

i in

v

to refer to a component of

v

$$

v

$,$ because

v

$$

v

is not a set. Instead, use

v

i

v

i

$$

to refer to the

i

ith component of

v

$$

v

$.$ What set would

i

i belong to in this case?