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Joelly
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MHB How to Choose the Right Running Shoes for You
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The discussion begins with a reference to abstract algebra concepts, specifically isomorphism, which involves a function between two groups that is one-to-one, onto, and preserves operations. The groups in question are defined as multiples of 2 (G) and multiples of 3 (H). The proposed function f(2n) = 3n is examined for its validity as an isomorphism. The key points focus on whether this function meets the criteria of being one-to-one, onto, and preserving the operation. Ultimately, the discussion highlights the mathematical relationships between these groups through the lens of isomorphic functions.
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HOI
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Are you taking an "Abstract Algebra". If so you should know what "isomorphic" means- that there is a function from one group to the other, f: G->H, that
1) it is "one-to-one"- if f(x)= f(y) the x= y.
2) it is "onto"- for every y in H there exist x in G such that f(x)= y.
3) it "preserves the operation"- f(x+ y)= f(x)+f(y).
G is defined as multiples of 2 and H is defined as multiples of 3. What about f(2n)= 3n?
1) it is "one-to-one"- if f(x)= f(y) the x= y.
2) it is "onto"- for every y in H there exist x in G such that f(x)= y.
3) it "preserves the operation"- f(x+ y)= f(x)+f(y).
G is defined as multiples of 2 and H is defined as multiples of 3. What about f(2n)= 3n?
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