Prove that the set S e2x x epsilon N is denumerable What is

Prove that the set S = e^2x: x epsilon N is denumerable. What is the cardinality of S?

Solution

An infinite set is denumerable if it is equivalent to the set of natural numbers. Here the given set is S = e^2x; x€N

The given set S = { e², e³, e⁴, ............} which is an infinite set,

Now e² = 1 + 2 + 2²/2! + 2³/3! +................. which is equivelent to the set of natural numbers

Similarly e³, e⁴, ... are also equivelent and therefore

the set S is denumerable and the cardinality of the set is infinite.