Assume that the integers have the following property: for any a,b element of Z(i

Question

Assume that the integers have the following property: for any a,b element of Z(integers) , ab=0 implies that either a=0 or b=0. This is reasonable to assume without much proof since it actually follows almost immediately from the way integers are defined. Using this fact as an axiom, prove that for any a,b,c element of Z, if ac=bc and c does not equal 0, then a=b. Note: this is proving that you can "divide" out c to get a=b as long as c does not equal 0. Since we do not want our reasoning to be circular, you cannot use cancellation or fractions in your proof.

Explanation / Answer

given ac = bc

=> ac - bc =0

=> (a-b)c = 0

consider a-b = p

=> pc = 0

since c != 0

p = 0

=> a-b = 0

a=b

HENCE PROVED

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