Boolean algebra, my first experience.
Today I returned a bit too late from FIITJEE considering that I had to give the evaluation forms to my teachers. After returning home, I picked up a book titled “An Unusual Algebra” by I.M. Yaglom. It is an excellent work that introduces Boolean algebra. I have finished half the book. Here is what I learned:
2 + 3 = 5
3 + 4 = 7
If we have sets like A, B and C and if we define addition to be union, and multiplication to be intersection, then we have the following properties associated with the operations addition and multiplication:
1. Commutative property:
A + B = B + A or A + C = C + A or B + C = C + B
AB = BA or AC = CA or BC = CB
2. Associative propery:
(A + B) + C = A + (B + C)
(AB)C = A(BC)
3. Distributive property:
(A + B)C = AC + BC
(A + C)(B + C) = AB + C
4. Idempotent property:
AA = A, BB = B and CC = C
A + A = A, B + B = B and C + C = C
So we go on to state that the operation “addition” and “multiplication” are to have the above properties and if we go on to apply this operation “addition” to a set of numbers {0, 1}, then we have the following:
0 + 1 = 1
0 + 0 = 0
1 + 1 = 1
1 + 0 = 1
Now these satisfy the properties stated above. There’s Boolean algebra in a nutshell.
I was then musing that those properties that we stated for sets form the peoperties for operations in Boolean algebra. However I did find a catch in that. We have what is known as the Identity element for addition and multiplication, 0 and 1 respectively. But there is no such set X such that X + A = A or XA = A. If there were such a set, it would be the superset of every set. There you go.
I need to learn a bit more. I will be posting more about this book here. Till then, goodbye
6 comments
I think you are misunderstanding some concepts here.. You assumed A, B and C to be sets, and then performed an addition operation with them. For example, A+A=A, because an addition here is just a union in fact. But then you came to the set {0, 1} and started to make the same operations (which you demonstrated on sets) with the individual elements of this particular set - with elements 0 and 1. This can’t be done in this way - you can’t do the same with elements, what you can do with sets.. So, for example, 1+1=0 in Boolean algebra, and it os not like it is with sets, when A+A=A..
Well the properties of addition in boolean algebras I suppose remain the same? I am not trying to state that union of sets = addition of numbers in boolean algebra. Do correct me if I am wrong.
Well, the addition in Boolean algebra is not like that you mentinoed above, it is as follows:
0+0=0
0+1=1
1+0=1
1+1=0
It can be perceived as addition by module 2 (so, for example, when adding 1 and 1, you get 2, but 2 is equal to zero by module 2 (that is, you get zero remainder when dividing the sum by 2))
Hi, i guess both are right but you have to take into account the relation you operating on your operands..
1 R 1 = 0, where ‘R’ is a XOR relation, or additon modulo 2 as edgarsr points out..
1 R 1 = 1, when ‘R’ is a OR or a disjunctive relation…
I guess the choice of the symbol is the confusing aspect here, if you are choosing addition mod 2, you are better off choosing ‘?’ instead of ‘+’..IMO thats the convention..
Hi, i guess both are right but you have to take into account the relation you operating on your operands..
1 R 1 = 0, where ‘R’ is a XOR relation, or additon modulo 2 as edgarsr points out..
1 R 1 = 1, when ‘R’ is a OR or a disjunctive relation…
I guess the choice of the symbol is the confusing aspect here, if you are choosing addition mod 2, you are better off choosing ‘?’ instead of ‘+’..AFAIK thats the convention..
very interesting, but I don’t agree with you
Idetrorce
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