Vocabulary in math logic

sara15

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if we let that V_gp be the vocabulary {+,0} where + is abinary function and 0 is a constant . we use the notation x+y to denote the term +(x,y) . Let gama be the conjunction of the following V-sentence: for all x the exist y for all z (x+(y+z)=(x+y)+z)
for all x (( x+0=x) conjunction ( o+x=x))
for all x ( there exists y (x+y=0) conjunction the exist z (z+x=0))
how can I give a model of gama that does not model the sentence for all x for all y((x+y)=(y+x)).
 
Should your first sentence have for all y (instead of exists y)? If so, then that would be associativity. Then those 3 sentences are the group axioms. So you just need a noncommutative group.
 
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Should your first sentence have for all y (instead of exists y)? If so, then that would be associativity. Then those 3 sentences are the group axioms. So you just need a noncommutative group.
?
 
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