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In which of following stuctures is valid implication $x\cdot y=1\implies x=1$?

a) $(\mathbb{N}, *)$

b) $(\mathbb{Z}, *)$

c) $(\mathbb{Q}, *)$

d) $(\mathbb{C}, *)$

Solution is a), I can prove it by using simple example for all of four sets, but I need more general explanation.

  • 2
    You are dealing with a multiplicative commutative monoid; $xy=1$ implies $x=1$ if and only if the only the only invertible is the identity element.2012-03-14
  • 0
    The tags were actually correct to begin with. This is a question about structures of predicate calculus, and logical implications.2012-03-14

3 Answers 3