2

Determine whether these structures are is isomorphic ? ,

if they are isomorphic Show the isomorphism function and proved

if they are no isomorphic proved by an appropriate verse

A. $M_2= \langle(1,\infty),+\rangle$ , $M_1 = \langle(0,\infty),+\rangle$

attempt:

if i want to show isomphism in A. or B. i need to show .

for all sign of function $f$

1.$h(F^{M}(a_1,a_2,...a_n))=F^N(h(a_1),h(a_2),...(h(a_n))$

2.Injective function

3.Surjective function

if i try to prove isomorphic in B. between $M_2$ , $M_1$ i dont know how to prove them with in open segment

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1 Answers1

3

Question A

The sentence $\phi \equiv \forall x \exists y(y+y=x)$ is satisfied in $M_1$ but not $M_2$. Therefore the two structures can't be isomorphic.

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  • $0\not\in(0,+\infty)$ –  Dec 11 '20 at 13:49
  • @StinkingBishop That is a good comment... Thanks! – mathcounterexamples.net Dec 11 '20 at 13:50
  • sorry some 1 edit my post its cannot be + in question B its a $\cdot$ in $M_1$ and $M_2$ .. –  Dec 11 '20 at 17:02
  • @Emily.d You were the one editing the post. Then I suggest that you take only point A in this question and open a new question for point B. Answering to moving question is really not pleasant! – mathcounterexamples.net Dec 11 '20 at 17:05
  • @mathcounterexamples.net i swear i didnt notice about that . i am really sorry !!..

    about question A you can please explain why $\phi \equiv \forall x \exists y(y+y=x)$ ?

     –  Dec 11 '20 at 17:07
  • @mathcounterexamples.net in this $ \forall x \exists y(y+y=x)$ in option A its see its isomphism –  Dec 11 '20 at 17:10
  • It is impossible to find an isomorphism for case A. An isomorphism preserves validity of sentences and $\phi$ is satisfied in $M_1$ and not in $M_2$. And again please move question B in another question as you edited it. – mathcounterexamples.net Dec 11 '20 at 17:16
  • @mathcounterexamples.net please you can explain the preserves validity of sentences ? and how its impossible that $\phi \equiv $ satisfied $M_1$ and not in $M_2$ because $M_1$ are numbers in range ( .. ) and $M_2$ ( .. ) so how the $\phi$ come to take a role . –  Dec 11 '20 at 17:44
  • I imagine that you're following (or reading) an introductory course in logic / model theory. You should refer to it as what I use is explained there. The $\phi$ I used is just saying for all element $x$ it exists an element that is half of $x$. This is true in $M_1$ but not in $M_2$. – mathcounterexamples.net Dec 11 '20 at 17:47
  • Let us continue this discussion in chat. –  Dec 11 '20 at 19:59