Are k, μ, v infinite cardinal numbers? Because otherwise, 1 is obviously not true, unless there is a typo there.
I need to prove the following rules on cardinals:
Let κ, μ, ν be cardinal numbers. Prove:
1. κ^(μ + ν) = κ^μ·κ^ν.
2. κ^μ·ν = (κ^μ)^ν.
How do I start proving such thing? It looks so obvious, but I'm sure there has to be a formal way to prove it.
Please help me!
Cardinal power is defined as , that is cardinality of the set of mappings from to .
So to prove for example (2) it suffices to show that has the same cardinality as .
For an arbitrary map and an element let us define map like this: for .
Then, relation is an injective mapping from to . If , then is preimage of under mapping . So is injective mapping from onto .
main idea is to construct a bijection between two sets. Firstly make sure you're ok with these series of equalities:
They justify that all you need to do is to show that . And that is done by the construction of the relation in my previous post and contemplating that is really a bijection.