A smal problem with boolean algebra

In summary, the conversation discusses the correctness of the equation f=xz +x'z', which is equivalent to (x xor z)'. De Morgan's law is used to prove this equivalence, and it also explains why (x xor y) xor (x xor z)' = (x xor y xor x xor z)'. The person also mentions the need to prove this equation using algebraic manipulations or by listing all possible values of x, y, and z.
  • #1
gipc
69
0
There's something I can't seem to figure out.

say we have the function
f=xz +x'z'

why is this equals (x xor z)' ?
i thought using De-Morgan's law we shall get

(x xor z)'=(x'z+z'x)'=(xz'zx')=0

then why does (x xor z)'=xz+x'z' ?


and one more small thing,
say we have
(x xor y) xor (x xor z)'
why does (x xor y) xor (x xor z)' = (x xor y xor x xor z)'

how did they come to this term?
 
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  • #2
De Morgan's law also tells you (ab)' = a' + b', so

(x xor z)' = (x'z+z'x)' = (x'z)'(z'x)' = ...

I'll let you finish it.
 
  • #3
thanks so much i feel so stupid for missing the obvious.

one more thing though,
why does the following equation stand?

(x XOR y) XOR (x XOR z)' = (x XOR y XOR x XOR z)'
 
  • #4
That's kind of like asking why a trig identity is true. It's true because it's a consequence of how things are defined. You just have to prove it by using algebraic manipulations or just listing all possible values of x, y, and z and showing the two sides give the same results.
 
  • #5



It seems like you are having trouble understanding the concept of xor (exclusive OR) in boolean algebra. Let's break it down step by step:

1. First, let's look at the function f = xz + x'z'. This function can also be written as f = xz + x'z' + 0. This is because 0 is the identity element for the boolean addition operation, which means that adding 0 to any boolean expression does not change its value. So, we can rewrite f as f = xz + x'z' + 0.

2. Now, let's use De-Morgan's law to simplify this expression. De-Morgan's law states that (A + B)' = A'B'. Applying this law to our expression, we get f = (xz)'(x'z')' + 0. Using De-Morgan's law again, we get f = (x' + z')(x + z) + 0. This simplifies to f = x'z' + xz + xz' + z'z + 0.

3. Notice that xz + xz' + z'z is equal to 0, because z and z' are complements of each other, meaning that they both can't be 1 at the same time. So, we can rewrite f as f = x'z' + xz + 0. This is where xor comes into play. Xor is the boolean operation that means "either A or B, but not both." So, in this case, we can rewrite f as f = x xor z.

4. Now, let's look at the expression (x xor y) xor (x xor z)'. This can also be written as (x xor y) xor (x xor z) + 0. Applying De-Morgan's law, we get (x' + y')(x' + z') + 0. Simplifying this, we get (x'x' + x'z' + x'y' + y'z') + 0. Again, notice that x'x' + x'y' + y'z' is equal to 0. So, we are left with (x'z' + x'z') + 0. This simplifies to (x' xor z') xor 0, which is equal to x' xor z'.

 

Related to A smal problem with boolean algebra

1. What is boolean algebra?

Boolean algebra is a branch of mathematics that deals with logical operations and variables. It is used to analyze and simplify logical expressions, and it serves as the foundation for digital electronics and computer programming.

2. What is a small problem with boolean algebra?

A common problem with boolean algebra is the confusion between the logical operators "and" and "or". "And" represents the intersection of two sets, while "or" represents the union of two sets. This can lead to errors in logical reasoning if not understood correctly.

3. How is boolean algebra used in science?

Boolean algebra is used in scientific research and experiments to analyze and interpret data. It is particularly useful in fields such as computer science, genetics, and physics to model complex systems and make predictions based on logical relationships.

4. What are the basic rules of boolean algebra?

The basic rules of boolean algebra include the commutative property (A OR B = B OR A), the associative property (A AND (B AND C) = (A AND B) AND C), and the distributive property (A AND (B OR C) = (A AND B) OR (A AND C)). Other rules include the identity laws, complement laws, and De Morgan's laws.

5. How can boolean algebra be applied in everyday life?

Boolean algebra can be applied in everyday life in various ways, such as in decision-making processes, problem-solving, and critical thinking. It can also be used in setting up and simplifying logical statements, organizing data, and creating algorithms for computer programs.

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