# 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

 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