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
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
what is answer c+c`
We must be careful when performing mathematical operations on sets rather than natural numbers. The similarties of properties is not present. The sameness which is present in the properties of addition and multiplication is not present. For example, adding 2+2 gives the result 4, but multiplying 2*2 also gives 4. The same thing does not hold true when we consider another number 3. The results are 6, and 9 respectively which denote different quantities. The reasoning should be present to give correct answer.
The same case obtains with the set thoery. There is an identify set which when multilpied retains the same result. Morevoer, the concepts of union, intersection etc partition the existing subsets. The logic of 1 or o are binary operations representing presence or absence of signal which is a result of an event taking place.