3
votes
2answers
92 views

What is the meaning of $n\in \aleph$

Using mathematical induction, prove that, for each $n\in \aleph$ $$n<3^n$$ Dear all, what is the meaning of "$n\in \aleph$" . How to substitute it and prove that? please give me one step ...
3
votes
4answers
315 views

Proving that there are infinite cardinal numbers >$\mathfrak{c}$

I was reading Simmons' book and he states that there are infinite cardinal numbers > $\mathfrak{c}$ where $\mathfrak{c}$ denotes the number of Real Numbers. For this, he states that we can construct ...
1
vote
1answer
136 views

Cardinality and ordinary mathematical induction

How do I approach this problem using ordinary mathematical induction? Notation: If A and B are sets then we will say they are the same cardinality and write $A\approx B$ if there is a one-to-one and ...