ok im confused on how to start with this
we had previously proved that A is a denumerable set and x is an element not in A, then A union {x} is also deumerable.
Prove that if S = {x1, x2, ...,xn} is a set of n elements, none of which are in A, then A union S is denumerable
Since is countable it follows that is countable and it follows that is countable assuming that is finite. The reason for the finiteness of is that any finite set is countable since if you have a set with elements you can easily find a bijection with the th natural number. If were infinite then we could have that or any other uncountable set, the only way to guarantee countability is finitness. In fact, it is true that if both are countable then so is regardless of their individual cardinalities, but since was not given as either countable or uncountable we may not use this argument.
EDIT: I completely missed you post Jhevon, my apologies.
in set theory, natural numbers are defined in terms of sets. zero is the null set, and every natural number after that is defined as the set containing all previous sets, so for instance, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, ...
so that if we have a set with n elements, we can find a bijection between that set and the nth natural number, since the natural number n is itself a set with n elements. thus we can do with finite sets and natural numbers the same thing we can do with infinite countable sets (a.k.a denumerable sets) and the set
yes, i mentioned that
indeed. i was merely explaining where Mathstud was coming from. i believe he was thinking of natural numbers in this context.but this is not the only way to define the natural numbers, some of us prefer the Peano axioms, so in a general setting we should avoid terminology that depends on a particular means of setting up the Naturals.
CB
of course, i do not think that this is how the OP would have the natural numbers defined in his/her course. which is why i suggested induction in the first place, to avoid this issue altogether.
zero is the null set
There is something disturbing me above, why wouldn't n be finite ? ^^'The reason for the finiteness of n is that any finite set is countable
If one defines a set of n elements, doesn't it mean it's finite ?
Oh right, again, it may be something related to the definition of natural numbers...
Hows this?
We are given that a set is countable, as well as the set . Now let us prove that for finite that is finite. To do this we use induction:
It is given that the base case, , is true.
Now let us restate our inductve hypothesis: If is finite and countable then is countable.
We left to prove that if is countable then so is . To do this we first observe that . Now if is countably infinite this is obviously true because the cardinality of would be . Now suppose that is finite with cardinality . Then by the countability of we may find a bijection between the th natural number and . Let's call this bijection . Now it is clear then that if we define the cardinality of as that we can construct the following function: which is bijective. Thus we have shown that there is a bijection between the th natural number and , thus it is countable.
Note: The neccessity of the finiteness of is that if were infinite then could be any number of uncountable sets such as the irrationals or the reals. The only way to guarantee the coutability of is to restrict its cardinality.