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Could you please help me with some questions.

1.26

Prove that there is a mapping from a set to itself that is one-to one but not onto iff there is a mapping from the set to itself that is onto but not one-to -one.

3.22

how many diff. commutative operations are there on a 1 element set?...n element set?

5.22

Prove that G is a group, a is in G, and a * b = b for some b in G, then a must be the identity of G.

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1.26

-> Suppose f:S->S is 1-1 but not onto. Define a mapping g:S->S as follows: g(x)=y if f(y)=x for some y in S and g(x)=some fixed s in S if such an element y does not exist. Now use the assumptions to show that g is onto but not 1-1.

[to help you understand the above here is an example of how g is constructed given a specific f. Suppose f:{all positive integers}->{all positive integers} is defined by f(n)=n+1. Then f is 1-1 but not onto (it is not onto since 1 is not hit by any element via f). Now the mapping g above is definied to do just the opposite of f for the numbers 2,3,...., (g(n)=n-1) and g(1) is any number (say 1).]

<- part is similar.

3.22. 1 on a set with 1 element.

2^{3} for 2 elements.

...

2^{nxn-(nxn-n)/2} for n elements.

(why ? because you can put whatever you want on the main diagonal and above the main diagonal in the Cayley table; the rest of the Cayley table is determined by the commutativity.)

5.22. Suppose a*b=b; then (a*b)*b^{-1}=b*b^{-1}; so a*(b*b^{-1})=e; so a=e.