I am taking a digital logic course as the first class required to start my CS degree. And, maybe it's just me, but my professor is very little help. All he seems to offer is lectures is the formula(which is also in the book) and no examples.
Anyway, point being, I would really appreciate any help if you can offer it.
At the moment we are studying Boolean algebra. The theorems are pretty confusing to apply to actual simplification problems (they give you something like "a + ab = a" as an example but then the problem at the end of the chapter is like "xz + wxyz".
So, if you understand Boolean Algebra or think you can help please message me or post in this topic and if you want I can put one of the problems I am having trouble with into this thread. (I'm not sure how to do an overline with text, maybe I could just take a picture?)
Ok, this one I think I did correctly(note that any variable with ' after it should be read as a complement [for example: a' is the complement of a]):
f(w,x,y,z)= x + xyz + x'yz + wx + w'x + x'y

 Using Theorem 6(a)  [ab + ab' = a]
 I made xyz + x'yz = yz
V = x + yz + wx + w'x + x'y

 Using T6(a) again
 I made wx + w'x = x
V = x + yz + x + xy

 Using T1(a)  [a + a = a]
 I made x + x = x
V = x + yz + x'y

 Using T5(a)  [a + a'b = a + b]
 I made x + x'y = x + y
V = x + yz + y

 Using T4(a)  [a + ab = a]
 I made y + yz = y
V = x + y (This is my final result)
This second one I have no idea at all:
f(x,y,z) = y'z(z' + z'x) + (x' + z')(x'y + x'z)

 The only thing I can think to do is to do is: z' + z'x = z' T4(a)
 but that doesn't really help me out
V = y'z(z') + (x' + z')(x'y + x'z)
If you need I can type out all the theorems, there are 9 of them each with two parts ( a and b )
and I think it is better to not factor y out and that way you can eliminate both x' and z ( from what I understand)
Anyway, here is a list of all the Postulates and theorems:
Postulate 1. Definition A Boolean algebra is a closed algebraic system containing a set K of two or more elements and the two operators . and +; alternatively, for every a and b in set K, a . b belongs to K and a + b belongs to K (+ is called OR and . is called AND).
Postulate 2. Existence of 1 and 0 elements
(a) a + 0 = a
(b ) a . 1 = a
Postulate 3. Commutativity of the + and . operations
(a) a + b = b + a
(b ) a . b = b . a
Postulate 4. Associativity of the + and . operations
(a) a + (b + c) = (a + b ) + c
(b ) a . (b . c) = (a . b ) . c
Postulate 5. Distributvity of + over . and . over +
(a) a + (b . c) = (a + b ) . (a + c)
(b ) a . (b + c) = (a . b ) + (a . c)
Postulate 6. Existence of the complement
(a) a + a' = 1
(b ) a . a' = 0
Duality
ex: a + (bc) = (a + b )(a + c)
Theorem 1. Idempotency
(a) a + a = a
(b ) a . a = a
Theorem 2. Null elements for + and . operators
(a) a + 1 = 1
(b ) a . 0 = 0
Theorem 3. Involution
a'' = a ('' the complement of a complement)
Theorem 4. Absorption
(a) a + ab = a
(b ) a(a + b ) = a
Theorem 5.
(a) a + a'b = a + b
(b ) a(a' + b ) = ab
Theorem 6.
(a) ab + ab' = a
(b ) (a + b )(a + b') = a
Theorem 7.
(a) ab + ab'c = ab + ac
(b ) (a + b )(a + b' + c) = (a + b )(a + c)
Theorem 8. DeMorgan's theorem( going to use *___* to represent the entire equation as a complement)
(a) *a + b* = a' . b'
(b ) *a . b* = a' + b'
Theorem 9. Consensus
(a) ab + a'c + bc = ab + a'c
(b ) (a + b )(a' + c)(b + c) = (a + b )(a' + c)
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Anyway, point being, I would really appreciate any help if you can offer it.
At the moment we are studying Boolean algebra. The theorems are pretty confusing to apply to actual simplification problems (they give you something like "a + ab = a" as an example but then the problem at the end of the chapter is like "xz + wxyz".
So, if you understand Boolean Algebra or think you can help please message me or post in this topic and if you want I can put one of the problems I am having trouble with into this thread. (I'm not sure how to do an overline with text, maybe I could just take a picture?)
f(w,x,y,z)= x + xyz + x'yz + wx + w'x + x'y

 Using Theorem 6(a)  [ab + ab' = a]
 I made xyz + x'yz = yz
V
= x + yz + wx + w'x + x'y

 Using T6(a) again
 I made wx + w'x = x
V
= x + yz + x + xy

 Using T1(a)  [a + a = a]
 I made x + x = x
V
= x + yz + x'y

 Using T5(a)  [a + a'b = a + b]
 I made x + x'y = x + y
V
= x + yz + y

 Using T4(a)  [a + ab = a]
 I made y + yz = y
V
= x + y (This is my final result)
This second one I have no idea at all:
f(x,y,z) = y'z(z' + z'x) + (x' + z')(x'y + x'z)

 The only thing I can think to do is to do is: z' + z'x = z' T4(a)
 but that doesn't really help me out
V
= y'z(z') + (x' + z')(x'y + x'z)
If you need I can type out all the theorems, there are 9 of them each with two parts ( a and b )
and I think it is better to not factor y out and that way you can eliminate both x' and z ( from what I understand)
Anyway, here is a list of all the Postulates and theorems:
Postulate 1. Definition A Boolean algebra is a closed algebraic system containing a set K of two or more elements and the two operators . and +; alternatively, for every a and b in set K, a . b belongs to K and a + b belongs to K (+ is called OR and . is called AND).
Postulate 2. Existence of 1 and 0 elements
(a) a + 0 = a
(b ) a . 1 = a
Postulate 3. Commutativity of the + and . operations
(a) a + b = b + a
(b ) a . b = b . a
Postulate 4. Associativity of the + and . operations
(a) a + (b + c) = (a + b ) + c
(b ) a . (b . c) = (a . b ) . c
Postulate 5. Distributvity of + over . and . over +
(a) a + (b . c) = (a + b ) . (a + c)
(b ) a . (b + c) = (a . b ) + (a . c)
Postulate 6. Existence of the complement
(a) a + a' = 1
(b ) a . a' = 0
Duality
ex: a + (bc) = (a + b )(a + c)
Theorem 1. Idempotency
(a) a + a = a
(b ) a . a = a
Theorem 2. Null elements for + and . operators
(a) a + 1 = 1
(b ) a . 0 = 0
Theorem 3. Involution
a'' = a ('' the complement of a complement)
Theorem 4. Absorption
(a) a + ab = a
(b ) a(a + b ) = a
Theorem 5.
(a) a + a'b = a + b
(b ) a(a' + b ) = ab
Theorem 6.
(a) ab + ab' = a
(b ) (a + b )(a + b') = a
Theorem 7.
(a) ab + ab'c = ab + ac
(b ) (a + b )(a + b' + c) = (a + b )(a + c)
Theorem 8. DeMorgan's theorem( going to use *___* to represent the entire equation as a complement)
(a) *a + b* = a' . b'
(b ) *a . b* = a' + b'
Theorem 9. Consensus
(a) ab + a'c + bc = ab + a'c
(b ) (a + b )(a' + c)(b + c) = (a + b )(a' + c)