- #1
chobo2
- 2
- 0
I would like you guys to check over my Boolean Algebra equations to see if I did them right.
My teacher uses the same symbols as the computer science books
Legend:
[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]=[/URL] XOR
XOR = a[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]b[/URL] = a * b' + a' * B
+ = OR
* = And
' = Not
a) X[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]1[/URL] = X
proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] + 1 = Def
X * 1' + X' * 1 = P5a
X * 0 + X' 1 = T9a, T10a
0 + X' = T9b, L6b
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 ≠ X
b) X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = X’
proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = Def
X* 1' + X' * 1 = P5a
X * 0 + X' * 1 = T9a,T10a
0 + X’ = L6b, T9b
X' =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = X’
c)
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = Def
X * 0' + X' * 0 = P5b
X * 1 + X' * 0 = T10a
X + X' * 0 = T9a
X + 0 = T9b
X=
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = X
d)
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] ) = X'
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = Def
X * 0' + X' * 0 = P5b
X * 1 + X' * 0 = T10a
X + X' * 0 = T9a
X + 0 = T9b
X =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 ≠ X'
e)
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' = 0
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' = Def
X * X'' + X' * X' = T13a
X * X + X' * X' = T11a
X + X’ * X' = T12a
X + 0 = T9b
X =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' ≠ 0
--------------------------------------------------------
Postulates:
P1a X = 1 if X ≠ 0 ----- P1b X = 0 if X ≠ 1
P2a 0*0 = 0 ----- P2b 0+0 = 0
P3a 1*1 = 1 ---- P3b 1+1 = 1
P4a 1*0 = 0 ----- P4b 1+0 = 1
P5a 1’ = 0 ----- P5b 0’ = 1
Algebraic Laws
Commutative laws
L6a x*y = y*x ----- L6b x+y = y+x
Associative laws
L7a x*(y*z) = (x*y)*z ----- L7b x+(y+z) = (x+y)+z
Distributive laws
L8a x*(y+z) = x*y+x*z ----- L8b x+y*z = (x+y)*(x+z)
Thermos
T9a x*0 = 0 ------ T9b x+0 = 0
T10a x*1 = x ------ T10b x+1 = 1
T11a x*x = x ------ T11b x+x = x
T12a x*x’ = 0 ------ T12b x+x’ = 1
T13a x’’ = x ------ T13b x = x’’
Absorption Theorems
T14a x+x*y = x -------- T14b x*(x+y)=x
T14c X*(x’ + y) = x*y ------- T14d x+x’*y= x+y
De Morgan’s Theorems
T15a (x*y*z)’ = x’ + y’ + z’
T15b (x+y+z)’ = x’ * y’ * z’
My teacher uses the same symbols as the computer science books
Legend:
[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]=[/URL] XOR
XOR = a[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]b[/URL] = a * b' + a' * B
+ = OR
* = And
' = Not
a) X[PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL]1[/URL] = X
proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] + 1 = Def
X * 1' + X' * 1 = P5a
X * 0 + X' 1 = T9a, T10a
0 + X' = T9b, L6b
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 ≠ X
b) X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = X’
proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = Def
X* 1' + X' * 1 = P5a
X * 0 + X' * 1 = T9a,T10a
0 + X’ = L6b, T9b
X' =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 1 = X’
c)
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = Def
X * 0' + X' * 0 = P5b
X * 1 + X' * 0 = T10a
X + X' * 0 = T9a
X + 0 = T9b
X=
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = X
d)
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] ) = X'
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 = Def
X * 0' + X' * 0 = P5b
X * 1 + X' * 0 = T10a
X + X' * 0 = T9a
X + 0 = T9b
X =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] 0 ≠ X'
e)
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' = 0
Proof:
X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' = Def
X * X'' + X' * X' = T13a
X * X + X' * X' = T11a
X + X’ * X' = T12a
X + 0 = T9b
X =
Conclusion X [PLAIN][PLAIN][PLAIN][PLAIN][PLAIN][PLAIN]http://img218.imageshack.us/img218/2948/xnor5qv.jpg [/URL][/URL][/URL][/URL][/URL] X' ≠ 0
--------------------------------------------------------
Postulates:
P1a X = 1 if X ≠ 0 ----- P1b X = 0 if X ≠ 1
P2a 0*0 = 0 ----- P2b 0+0 = 0
P3a 1*1 = 1 ---- P3b 1+1 = 1
P4a 1*0 = 0 ----- P4b 1+0 = 1
P5a 1’ = 0 ----- P5b 0’ = 1
Algebraic Laws
Commutative laws
L6a x*y = y*x ----- L6b x+y = y+x
Associative laws
L7a x*(y*z) = (x*y)*z ----- L7b x+(y+z) = (x+y)+z
Distributive laws
L8a x*(y+z) = x*y+x*z ----- L8b x+y*z = (x+y)*(x+z)
Thermos
T9a x*0 = 0 ------ T9b x+0 = 0
T10a x*1 = x ------ T10b x+1 = 1
T11a x*x = x ------ T11b x+x = x
T12a x*x’ = 0 ------ T12b x+x’ = 1
T13a x’’ = x ------ T13b x = x’’
Absorption Theorems
T14a x+x*y = x -------- T14b x*(x+y)=x
T14c X*(x’ + y) = x*y ------- T14d x+x’*y= x+y
De Morgan’s Theorems
T15a (x*y*z)’ = x’ + y’ + z’
T15b (x+y+z)’ = x’ * y’ * z’
Last edited by a moderator: