can somebody give the definition of rigorous proof??

 

How for example one would write a rigorous proof for:

 

(-x)(-y) = xy for all x,y.

 

thank you

I am sure there is a very concise definition within proof theory, but a workable definition is a proof is a convincing argument that some statement is true. Convincing really needs to be understood within the framework you are using. That is, one would take as accepted certain theorems and lemmas in that branch of mathematics as given.

 

 

It is this notion of "convincing" you need to think about. What are you going to take as accepted and where are you going to start.

 

Most proof in practice are informal by the standards of proof theory and formal logic. I don't know much about there areas, also I don't think they would help you much.

I am sure there is a very concise definition within proof theory, but a workable definition is a proof is a convincing argument that some statement is true. Convincing really needs to be understood within the framework you are using. That is, one would take as accepted certain theorems and lemmas in that branch of mathematics as given.

 

 

It is this notion of "convincing" you need to think about. What are you going to take as accepted and where are you going to start.

 

Most proof in practice are informal by the standards of proof theory and formal logic. I don't know much about there areas, also I don't think they would help you much.

 

THANK YOU.

 

Would you consider the following as rigorous:

 

1) (-a)0 =0..............................................by using the theorem:[math]\forall x(x0=0)[/math]

 

2) b+(-b)=0.............................................by using the axiom:[math]\forall x(x+(-x)=0)[/math]

 

3)(-a)[b+(-b)] =0....................................by substituting (2) into (1)

 

4) (-a)[b+(-b)] = (-a)b +(-a)(-b)...............by using the axiom:[math]\forall x\forall y\forall z[x(y+z)=xy+xz][/math]

 

5) (-a)b+(-a)(-b)=0..................................by substituting (4) into (3)

 

6) 0b =0.....................................................by using the theorem:[math]\forall x(0x=0)[/math]

 

7) a+(-a)=0..............................................by using the axiom:[math]\forall x(x+(-x)=0)[/math]

 

8) [a+(-a)]b = 0........................................by substituting (7) into (6)

 

9) [a+(-a)]b = ab+(-a)b............................by using the axiom:[math]\forall x\forall y\forall z[(x+y)z = xz+yz][/math]

 

10) ab+(-a)b =0........................................by substituting (9) into (8)

 

11) ab+(-a)b = (-a)b+ab............................by using the axiom:[math]\forall x\forall y(x+y=y+x)[/math]

 

12) (-a)b+ab =0.........................................by substituting (11) into (10)

 

13) (-a)b+(-a)(-b)=(-a)b+ab......................by substituting (12) into (5)

 

14) (-a)(-b) = ab........................................by using the cancellation law:[math]\forall x\forall y\forall z( x+y=x+z\Longrightarrow y=z)[/math]