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Mathematical Thinking - ∃, there exists

by OLU
(UK)











































∃, there exists

   • true or false

           3m + 5n = 12

       when

           ∃m ∈ ℕ

           ∃n ∈ ℕ

Comments for Mathematical Thinking - ∃, there exists

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Oct 24, 2012
∃, there exists
by: Staff


Answer

Part I

∃m∈ℕ means there exists at least one “m” which is an element of the set of natural numbers

∃n ∈ℕ means there exists at least one “n” which is an element of the set of natural numbers


Can 3m + 5n = 12 ?

The answer to the question depends on your definition of what a natural number is.

ℕ = the set of all natural numbers

There is no universal agreement on the definition of a natural number.


The traditional definition of natural number 

does not include the number 0
:

ℕ (traditional definition) = positive integers

ℕ (traditional definition) = {1,2,3,4,…}


An alternative definition is that natural numbers
are the set of counting numbers which include
the number 0
:

ℕ (alternative definition) = non-negative integers

ℕ (alternative definition) = {0,1,2,3,4,…}


----------------------------


Oct 24, 2012
∃, there exists
by: Staff


----------------------------

Part II

Is it possible that 3m + 5n = 12 ?

Using the traditional definition for natural numbers, 

m and n can only be one of the following values:

= {1,2,3,4,5,6,7,8,9,10,11,12}

If m = 1 and n = 1, then

3m + 5n= ? = 3*1 + 5*1 = 8 ≠ 12

If m = 1 and n = 2, then

3m + 5n= ? = 3*1 + 5*2 = 13 ≠ 12

If m = 2 and n = 1, then

3m + 5n= ? = 3*2 + 5*1 = 11 ≠ 12

If m = 2 and n = 2, then

3m + 5n= ? = 3*2 + 5*2 = 16 ≠ 12

If m = 3 and n = 1, then

3m + 5n= ? = 3*3 + 5*1 = 14 ≠ 12


Using the alternative definition for natural numbers,
m and n can be one of the following values:

= {0,1,2,3,4,5,6,7,8,9,10,11,12}

If m = 0 and n = 1, then

3m + 5n = ? = 3*0 + 5*1 = 5 ≠ 12

If m = 1 and n = 0, then

3m + 5n = ? = 3*1 + 5*0 = 3 ≠ 12

If m = 2 and n = 0, then

3m + 5n = ? = 3*2 + 5*0 = 6 ≠ 12

If m = 0 and n = 2, then

3m + 5n = ? = 3*0 + 5*2 = 10 ≠ 12

If m = 3 and n = 0, then

3m + 5n = ? = 3*3 + 5*0 = 9 ≠ 12

If m = 0 and n = 2, then

3m + 5n = ? = 3*0 + 5*2 = 10 ≠ 12

If m = 4 and n = 0, then

3m + 5n = ? = 3*4 + 5*0 = 12



Final Answer:

           Using the traditional definition for a natural number, 3m + 5n ≠ 12

           Using the alternative definition for a natural number, 3m + 5n = 12 when m = 4 and n = 0




Thanks for writing.

Staff
www.solving-math-problems.com


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