- #1
MathematicalPhysicist
Gold Member
- 4,699
- 371
i have a few question, that i hope they will answered.
1) let w={0,1...,n,..}={0}UN, and let f:wxw->w such that the next requirements apply:
a) f(0,n)=n+1
b) f(m+1,0)=f(m,1)
c) f(m+1,n+1)=f(m,f(m+1,n).
i need to prove that for every n,m in w, the next statement applies:
f(m,n)<f(m,n+1).
(i tried with induction, but this was too much complicated for me to do).
2)let there be 2 sets, X and Y, prove with the help of AC that there exist a function from X onto Y iff |X|>=|Y|.
well for the second question, here what i did:
suppose, there exists f:X->Y which is onto Y, if we utilize AC then let's define P(Y)\{empty set} as a subset of X, then by AC f|P(Y)\{empty set} P{Y}\{EMPTY SET}->Y such that f|P(X)\{ES}(A) is in A for every A in P(Y)\{ES} which means Y is onto P(Y)\{ES} and therefore there is bijection between Y and a subset of X, therefore |X|>=|Y|, the second conditional here I am not sure hoe to prove, i will appreciate your help.
1) let w={0,1...,n,..}={0}UN, and let f:wxw->w such that the next requirements apply:
a) f(0,n)=n+1
b) f(m+1,0)=f(m,1)
c) f(m+1,n+1)=f(m,f(m+1,n).
i need to prove that for every n,m in w, the next statement applies:
f(m,n)<f(m,n+1).
(i tried with induction, but this was too much complicated for me to do).
2)let there be 2 sets, X and Y, prove with the help of AC that there exist a function from X onto Y iff |X|>=|Y|.
well for the second question, here what i did:
suppose, there exists f:X->Y which is onto Y, if we utilize AC then let's define P(Y)\{empty set} as a subset of X, then by AC f|P(Y)\{empty set} P{Y}\{EMPTY SET}->Y such that f|P(X)\{ES}(A) is in A for every A in P(Y)\{ES} which means Y is onto P(Y)\{ES} and therefore there is bijection between Y and a subset of X, therefore |X|>=|Y|, the second conditional here I am not sure hoe to prove, i will appreciate your help.