Question

A set X is said to be closed under multiplication if for every X1, X2 E X we have X1X2 E X. Let A be the union of all bounded subsets X CR that are closed under multiplication. Does inf(A) exist? If it does, find it.

Answer 1

Answer:

inf(A) does not exist.

Step-by-step explanation:

As per the question:

We need to prove that A is closed under multiplication,

If for every$X_{1}, X_{2}\in X$

$X_{1}X_{2}\in X$

Proof:

Suppose, x, y $\in A$

Since, both x and y are real numbers thus xy is also a real number.

Now, consider another set B such that:

B = {xy} has only a single element 'xy' and thus [B] is bounded.

Since, [A] represents the union of all the bounded sets, therefore,

$B\subset A$

⇒ xy $\in A$

Therefore, from x, y $\]in A$, we have xy $\]in A$.

Hence, set a is closed under multiplication.

Now, to prove whether inf(A) exist or not

Proof:

Let us assume that inf(A) exist and inf(A) = $\beta$

Thus $\beta$ is also a real number.

Let C be another set such that

C = { $\beta$ - 1}

Now, we know that C is a bounded set thus { $\beta$ - 1} is also an element of A

Also, we know:

inf(A) =  $\beta$

Therefore,

$n(A)\geq \beta$

But

$\beta - 1$ is an element of A and  $\beta - 1 \leq \beta$

This is contradictory, thus inf(A) does not exist.

Hence, proved.