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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beckden | High School Teacher | (Level 1) Educator

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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.

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sainib | College Teacher | (Level 1) eNoter

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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.

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