Boolean algebra, what is a "unary" operator '?

Question

Formally, a Boolean algebra is a mathematical system consisting of a set of elements, B={0,1}, together with two binary operations, (denoted by the symbols + and •) There is also a unary operator ′.

I understand the laws and the operations ( and, or, not). I just need help understanding this unary ' thing.

For instance : Z=( A ' • B ' ) '+ C ' ) '

Also : Z = ( A • B ' ) ' + B ) • C

- I thought it was the "not" symbol until i saw these. Please help me understand.

I understand what you're saying kickthecan61. The thing that got me confused is what McFate mentioned. I was fine until i found these not not's. Guess i ignore the extras?

Answer 1

"Unary" just means "operates on one thing only." Negation ("-x") is a unary operator that you often see in standard math.

"Not" is an example of a unary operator. It looks to me as if they mean + (or), * (and), and ' (not) for their operations.... though the parentheses are not balanced in your examples.

Z=( A ' • B ' ) '+ C ' ) '

I would read that as:

Z = not{ not[ not(a) AND not(b) ] or not(c) }

PS - You can't just "ignore the extras." All of those "nots" have meaning.

For example, "not[ not(a) AND not(b) ]" is shown in this table, along with intermediate calculated values:

a, b, a', b', (a' and b'), (a' and b')':
T, T, F, F, F, T
T, F, F, T, F, T
F, T, T, F, F, T
F, F, T, T, T, F

Note that "not( not(a) AND not(b) )" is exactly the same as "a OR b" (false only when a and b are both false).

Personally, I would simplify:

Z = ( ( A ' • B ' ) '+ C ' ) '

... as:

Z = ( (A + B)' • C )

Answer 2

You are correct ' is the symbol being used for NOT.

AND (*) and OR (+) are binary operators because they work with TWO things [operands] e.g. T * F gives F

NOT (') is a unary operator because it only needs ONE operand e.g. T' gives F

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Unary operators usually take precedence over binary operators. So with A * B' , B' is evaluated first.

With the problems like you gave as examples, its best to work from the inside of the parenthesis out e.g. in ( A • B ' ) ' + B ) • C do the A*B' first ... another example...
    (F' * T)' -> (T * T)' -> (T)' -> F

OR use some of the properties of Boolean algebra to rearrange things
    e.g. [distributing the not]    (A * B)' = A' + B'
    also (A * B')' = A' + B since B'' = B

Answer 3

A unary operator acts on one element (negation or "not" in Boolean algebra)
A binary operator acts on two elements (like + or *)