- #1
evagelos
- 315
- 0
I was asked to write a rigorous proof for the following theorem:
0x = 0 ,for all x.
Is the following rigorous proof correct??
1) 0x = 0x+0...........by using the axiom:for all ,a : a+0=a
2) x+(-x) = 0..........by using the axiom: for all ,a: a+(-a) = 0
3) 0x = 0x +(x+(-x)).........by substituting (2) into (1)
4) 0x+(x+(-x)) = (0x+x)+(-x)......by using the axiom:for all a,b,c:a+(b+c)=(a+b)+c
5) 0x = (0x+x)+(-x).........by substituting (4) into (3)
6) 0x+x = x+0x.........by using the axiom:for all a,b:a+b=b+a
7) 0x = (x+0x)+(-x).........by substituting (6) into (5)
8) 1x = x............by using the axiom:for all,a:1a = a
9) 0x = (1x+0x)+(-x).........by substituting (8) into (7)
10) 1x+0x = (1+0)x.........by using the axiom: for all a,b,c:(a+b)c= ac+bc
11) 0x = (1+0)x+(-x).........by substituting (10) into (9)
12) 1+0 = 1...........by using the axiom:for all,a:a+0=a
13) 0x = 1x+(-x).........by substituting (12) into (11)
14) 1x = x..........by using the axiom:for all,a:1a = a
15) 0x = x+(-x).........by substituting (14) into (13)
16) x+(-x) = 0.........by using the axiom:for all,a:a+(-a) = 0
17) 0x = 0..........by substituting (16) into (15)
Thanx ,any help will be wellcomed
0x = 0 ,for all x.
Is the following rigorous proof correct??
1) 0x = 0x+0...........by using the axiom:for all ,a : a+0=a
2) x+(-x) = 0..........by using the axiom: for all ,a: a+(-a) = 0
3) 0x = 0x +(x+(-x)).........by substituting (2) into (1)
4) 0x+(x+(-x)) = (0x+x)+(-x)......by using the axiom:for all a,b,c:a+(b+c)=(a+b)+c
5) 0x = (0x+x)+(-x).........by substituting (4) into (3)
6) 0x+x = x+0x.........by using the axiom:for all a,b:a+b=b+a
7) 0x = (x+0x)+(-x).........by substituting (6) into (5)
8) 1x = x............by using the axiom:for all,a:1a = a
9) 0x = (1x+0x)+(-x).........by substituting (8) into (7)
10) 1x+0x = (1+0)x.........by using the axiom: for all a,b,c:(a+b)c= ac+bc
11) 0x = (1+0)x+(-x).........by substituting (10) into (9)
12) 1+0 = 1...........by using the axiom:for all,a:a+0=a
13) 0x = 1x+(-x).........by substituting (12) into (11)
14) 1x = x..........by using the axiom:for all,a:1a = a
15) 0x = x+(-x).........by substituting (14) into (13)
16) x+(-x) = 0.........by using the axiom:for all,a:a+(-a) = 0
17) 0x = 0..........by substituting (16) into (15)
Thanx ,any help will be wellcomed