Is B={0,1,R,F,X} a Boolean Algebra?

[hime]

Homework Statement

is B={0,1,R,F,X) a boolean algebra? Use basic posulates/axioms to prove it.

R=Rising
F=Falling
X=Dont Care

Homework Equations

Reference: Boolean Identities Table

The Attempt at a Solution

it is boolean algebra. you can create and, or , not tables with it.

please help.

 

[Grep]

Did you try and make some logic tables for it? Doesn't seem so.

You'd end up with problems if you did. For example, what is 'not X'?

Or what is 'not F'? Surely it can't be R since if it's not falling, it could be steady. So not necessarily R, logically speaking.

What is '0 and R'?

I'm not sure what your prof is expecting here, exactly, but perhaps that will help. Would be good to know what axioms and postulates he has introduced.

To my mind, since boolean algebra could be described as "two-valued logic", it's obviously not true with 5 values that cannot be reduced to be equivalent to 2 values.

 

[hime]

AND | 0 1 R F X
---- |---------
---0 | 0 0 0 0 0
---1 | 0 1 R F X
---R | 0 R R X X
---F | 0 F X F X
---X | 0 X X X X

NOT
x |~x
------
0 |1
1 |0
R |F
F |R
X |X

so obviously R' =F and vice versa

 

[Grep]

Well, let's see. We have one identity that is:

A and ~A = 0

So we should get:

R and ~R = 0

However, by your truth tables, ~R = F, that means:

R and ~R = R and F = X

However, it must equal 0, not X.

 

[DeeNos]

Yes, B={0,1,R,F,X} can be considered a Boolean algebra. In order to prove this, we can use the basic postulates/axioms of Boolean algebra and show that they hold true for the given set B.

1. Closure Property: The set B is closed under the operations of AND, OR, and NOT. This means that when any two elements of B are combined using these operations, the result will still be an element of B. For example, 0 AND R = 0, which is an element of B.

2. Associative Property: The operations of AND and OR are associative in B, meaning that the order in which they are performed does not affect the result. For example, (1 OR F) OR X = 1 OR (F OR X) = 1, which are all elements of B.

3. Commutative Property: The operations of AND and OR are commutative in B, meaning that the order of the operands does not affect the result. For example, 1 AND R = R AND 1 = R, which are all elements of B.

4. Identity Property: The elements 0 and 1 act as identities for the operations of AND and OR, respectively. For example, 1 AND X = X, which is an element of B.

5. Complement Property: Each element in B has a complement, denoted by a bar over the element, such that when the element is combined with its complement using the operation of OR, the result is 1, and when combined using the operation of AND, the result is 0. For example, R' OR R = 1 and R' AND R = 0, both of which are elements of B.

Therefore, B={0,1,R,F,X} satisfies all the basic postulates/axioms of Boolean algebra, and hence can be considered a Boolean algebra.