Let  Which of the following is true about S? 

Concept:

A bijection is a one-to-one and onto mapping between two sets, meaning every element in one set

corresponds uniquely to an element in the other set, and vice versa.

Since  is an infinite and uncountable set, it has the same cardinality as  (the set of all real numbers).

Therefore, there exists a bijection between  and  . . . . .

Explanation:

Let's simplify this expression 

Now factoring gives

= 

The exact algebraic manipulation of this inequality is complex, but we can understand its

behavior based on . For large values of  , the numerator grows faster than the denominator, so the

expression eventually becomes greater than 22 for sufficiently large  .

For , there exist values of  that satisfy the inequality, and the inequality is true for sufficiently

large . This implies that  is an infinite set, and it has no upper bound.

Option 1: "S is empty" — This is false because the inequality holds for large .
 

Option 2: "There is a bijection between  and " — This is false .  is an infinite set, but it's uncountable

like , so it can't be in bijection with  (the natural numbers, which are countable).
 

Option 3: "There is a bijection between  and " — This is true. The set , being infinite and

uncountable for , has the same cardinality as , so there exists a bijection between  and 
Option 4: "There is a bijection between  and a non-empty finite set" — This is false.  is infinite.

The correct answer is Option 3).

$$\text{There is a bijection between } S \text{ and } \mathbb{N}.$$