The Zero product property is:
- if xy=0 then x=0 or y=0
- A student asks,
Using only the following laws prove that if mn=0, then either m=0 or n=0.
- i) if m and n are in D then m+n and mn are in D.
- ii) if m=p and n=q in D then m+n=p+q and mn=pq.
- iii) m+n = n+m
- iv) mn=nm
- v) m+(n+p)=(m+n)+p
- vi) m(np)=(mn)p
- vii) m(n+p) = mn + mp
- viii) if m =/= 0, then mn = mp ==> n=p.
The number 0 has the following properties. For each number m in D,
- ix) m+0 = m
- x) m.0 = 0
- xi) there exists a number -m such that m+(-m) = 0
The number 1 =/= 0 has the property:
- xii) m.1 = m for all numbers m in D.
- MathHelp replies,
These are the field properties minus the multiplicative inverse, and with an additional axiom viii that allows you to cancel a common nonzero factor. Axiom ii is superfluous, and just states what is meant by the use of symbols. Axiom x is also superfluous, as it can be proved from the other axioms.
- m.0 = m.(0+0) = m.0+m.0, and by adding (-m.0) to both sides, we get 0=m.0
Proof of zero product propertyEdit
We're being asked to prove the statement, if mn=0, then either m=0 or n=0. Let's prove instead the contrapositive, namely
While we're in a contrapositive mood, let's restate viii that way:
If we let p=0, and using the fact that m.0=0, then the statement of axiom viii becomes
Composing the implications, we get the statement we need, which is