If S is countable and nonempty, prove their exist a surjection g: N --> S
1 answer:
Answer with Step-by-step explanation:
We are given S be any set which is countable and nonempty.
We have to prove that their exist a surjection g:N
Surjection: It is also called onto function .When cardinality of domain set is greater than or equal to cardinality of range set then the function is onto
Cardinality of natural numbers set =( Aleph naught)
There are two cases
1.S is finite nonempty set
2.S is countably infinite set
1.When S is finite set and nonempty set
Then cardinality of set S is any constant number which is less than the cardinality of set of natura number
Therefore, their exist a surjection from N to S.
2.When S is countably infinite set and cardinality with aleph naught
Then cardinality of set S is equal to cardinality of set of natural .Therefore, their exist a surjection from N to S.
Hence, proved
You might be interested in
We need to solve for x. Let's try problem b:
Let us first combine line terms. 3x and -x as well as 1 and -7 can be combined. Let's do that:
Since this is true, your answer would be:
All real numbers
------------------------------------------------------------------
Let's solve for problem c:
Let's isolate x, so subtract 1 from both sides:
Since x can't have a coefficient, divide both sides by 3:
So, only the value of 14 would make this equation true.
------------------------------------------------------------------
Let's try problem d:
Let's get our whole numbers on the right side. Add 1 to both sides:
Subtract 4x from the right side on both sides:
Since this is not true, your answer would be:
No solution