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    Prove a semigroup is a group

    Homework Statement Prove, that a senigruop S is a group if and only if for every x in S there exists a unique x' in S so that xx'x=x. Homework Equations The Attempt at a Solution I am trying to prove that xx'=1S for all x in S. But all I can show is that xx' is in ES={x | x^2=x}...
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    Propositional logic

    Homework Statement Either A or B (names changed) stole the exam answers. Formalize these and check if this is a correct deduction: 1) If A didn't meet B for lunch, then B is guilty or A lives in the countryside 2) If B isn't guilty, then A didn't meet B for lunch and the incident happened...
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    Simple number theory, divisibility

    Can you help me with this problem: if a^2 divides b^2, show that a divides b. This was a homework question that I had a while ago and it was solved by using the fundamental theorem of arithmetic. I instead tried to solve it with proof by contradiction: a^2 divides b^2 implies a divides...