PEMBUKTIAN 3+3=6

sebelum melakukan pembuktian bahwa 3+3=6,  tentunya kita harus memahami terlebih dahulu aksioma yang berlaku pada bilangan asli (natural number) dan aksioma peano (peano axioms). berikut adalah aksioma tersebut:

Axiom 1. For every x N, x = x.

Axiom 2. For every x, y N, if x = y, then y = x.

Axiom 3. For every x, y, z N, if x = y and y = z, then x = z.

Axiom 4. For all x and y, if x N and x = y, then y N

Peano axioms

Axiom 5. 0 is a natural number. That is, 0 N.

Axiom 6. If x N, then S(x) N. That is, if x is a natural number, then so
its successor.

Axiom 7. For every natural number x N, S(x) = 0 is false.

Axiom 8. For all x, y N, if S(x) = S(y), then x = y.

Axiom 9. If V is an inductive set, then N V .

 

suksesor pada natural number dapat dinyatakan dalam

S(0) = 1, S(1) = 2, S(2) = 3, S(3) = 4 dan seterusnya.

DEFINISI PENJUMLAHAN “…+…”

  • a + 0 = a.
  • a + S(b) = S(a + b)

berdasarkan definisi di atas dapat kita tuliskan

1+1= 1+S(0)=S(1+0)=S(1)= 2.

Jadi 1+1=2

1+2= 1+S(1)=S(1+1)=S(2)= 3.

Jadi 1+2=3

1+3= 1+S(2)=S(1+2)=S(3)= 4

Jadi 1+3 =4, karena berlaku komutatif (pembuktiannya silakan dicari sendiri) maka 3+1=4

Maka

3+3= 3+S(2) = S(3+2)

=S(3+S(1)) = S( S(3+1)) = S(S(4)) =S(5)= 6

Jadi  3+3=6, Terbukti
🙂

 

 

 

Categories: Abstrac Algebra, Home, Mathematics, Problem Solving, Tak Berkategori | Tags: , , , , , | Leave a comment

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