Gale
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Homework Statement
(i) If (X,*) is a binary operation, show that the identity function
Id_X : X \rightarrow Xis an isomorphism.
(ii) Let (X_1, *_1) and (X_2, *_2) be two binary structures and let f : X_1 \rightarrow X_2 be an isomorphism of the binary structures. Show that f^-1 : X_2 \rightarrow X_1 is also an isomorphism.
(iii) Let (X_1, *_1), (X_2, *_2), (X_3, *_3) be three binary structures and
let f : X_1 \rightarrow X_2 and g : X_2 \rightarrow X_3 be isomorphisms of the binary structures. Show that g \circ f : X_1 \rightarrow X_3 is also an isomorphism.
(iv) Denote the statement that (X_1,*_1) and (X_2, *_2)are isomorphic by (X_1, *_1) \cong (X_2, *_2). Using the above, show that \cong is reflexive, symmetric and transitive.
Homework Equations
The Attempt at a Solution
Okay, so I'm a bit confused with how to work with isomorphisms and binary operations in general. I'm don't know how to approach the first half of the problem, so I can't really do the rest either. Am I supposed to choose elements from the set X and work my proofs from there, or is there some other approach I should be taking? Besides that, I'm not sure I entirely understand the more general premise of the problems:
Starting with i) I'm not sure why the identity function is only isomorphic when there exists a binary relation. I'm not very confident in my understanding, but it seems like the identity function would always be isomorphic?
ii) I'm not sure how to start a proof for this, but since f is an isomorphism, isn't it necessarily bijective so obviously it would have an inverse? I'm confused as to what the proof is supposed to prove?
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