# Prove: abc = 0 ---> a = 0 or b = 0 or c = 0Prove using field axioms and axioms of equality. Use the "statement" and "reason" format to come up with an answer.

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### 2 Answers

The zero product property:

If a*b = 0 then either a = 0 or b = 0.

Now we have a*b*c = 0

Using the associative property of multiplication

a*b*c = (a*b)*c = 0

So either a*b = 0 or c = 0

If a*b*c = 0 then a*b = 0 or c = 0

Using the zero product property

if a*b = 0 then a = 0 or b = 0 and this gives us the final result that

a*b*c = 0 then a = 0 or b = 0 or c = 0.

*The common definition of the word "axiom" is "self-evident truth." In mathematics, an axiom is a statement whose truth is either to be taken as self-evident or to be assumed. Starting from a system of axioms, one can make certain logical deductions that all take the following form: if the axioms hold, then something else also does. The truth of the axioms is not in dispute. If something satisfies the assumptions, the conclusions derived from them are guaranteed to *be valid. Therefore, the axiomatic method can derive a large body of theory from a small number of assumptions. Instead of going through all the checking over and over again, we only need to verify a few properties, thus achieving a significant savings of time and effort.

Multiplicative Property of Zero

For any number a, the product of a and zero is always zero.

For example, 4 · 0 = 0. Also, 0 · 4 = 0.

Assumption:

a · b = 0 if a = 0 or b = 0

Reason:

If 4 · 1 = 4

Then it follows that 4 · 0.1 = 0.4 and 0.1 · 0.1 = 0.01 therefore it follows as the two multipliers become smaller and smaller the resultant approaches zero.

0.01 · 0.001 = 0.000001 etc.

Similarly

If a · b = 0 if a = 0 or b = 0

Then b · c = 0 if a = 0 or c = 0

And a ·b · c = 0 if a = 0 , b = 0 and c = 0 Q.E.D.