ASSOCIATIVE OPERATOR

[Dr.Peterson]

So * can be any arbitrary operation .
And it is named " product"
The 5 ways that they have given keeping in mind that these are Just enough to show any type of arbitrary operation whether it will be Associative or non associative are just "Fully parenthesized" which is extremely important if operation is non associative!


The list is just fully parenthesized so that any non associative operation can be evaluated without any problem as the order in Which
operation will be performed via " parentheses" has been clearly described Which subexpression to evaluate in order

that's why my two forms are not validate in this list.

Correct. Their list covers all possible orders in which the operations can be executed, without regard to the fact that, if the operation is associative, they will all have the same value. Your additions (with fewer parentheses) did not fully specify the order explicitly; and additional parentheses could have been added which would not result in different orders.

 

[Saumyojit]

In a+b+c expression if I want to show associative by changing parentheses

Then, (a+b)+c=a+(b+c)

In a-b-c if I change parantheses then,

1st case:
(a-b)-c =a-(b-c)

2nd case:
(a-b)-c=a-(b +c) (this is maintaining the originality of the " a-b-c" )

Why the 1st Case of a-b-c is right compared to 2nd case even though it is not maintaining the originality of the original expression.

My reason:
arranging Parentheses just means Only arranging Parentheses without changing any sign of each term.
Right?
@Dr.Peterson
@JeffM

 

[Dr.Peterson]

Your first case is "right" in the sense that this is what it would mean IF subtraction, as an operation, were associative. The fact that it is not true shows that it is not associative.

Your second case is "right" in the sense that you have changed an operation in order to obtain a true equation. This is what one actually does in manipulating an expression when doing algebra. So this is really the more important thing to know! And in knowing to do this, you are showing that you know subtraction is not associative.

Your question reminds me of this one from 12 years ago: Are Subtraction and Division Commutative and Associative Operations?

 

[JeffM]

In a+b+c expression if I want to show associative by changing parentheses

Then, (a+b)+c=a+(b+c)

In a-b-c if I change parantheses then,

1st case:
(a-b)-c =a-(b-c)

2nd case:
(a-b)-c=a-(b +c) (this is maintaining the originality of the " a-b-c" )

Why the 1st Case of a-b-c is right compared to 2nd case even though it is not maintaining the originality of the original expression.

My reason:
arranging Parentheses just means Only arranging Parentheses without changing any sign of each term.
Right?
@Dr.Peterson
@JeffM

Parentheses specifically (grouping symbols generally) tell us which operations to do first.

(17 - 9) - 3 = 8 - 3 = 5.

17 - (9 - 3) = 17 - 6 = 11.

So it is obviously INCORRECT to say that (a - b) - c = a - (b - c). That is generally untrue. Why do you say the 1st case is correct when it obviously is not?

(17 - 9) - 3 = 8 - 3 = 5.

17 - (9 + 3) = 17 - 12 = 5.

That is not a proof that (a - b) - c = a - (b + c) in every case, but I suggest that you experiment with a few hundred triplets of numbers. That experiment might convince you that it is reasonable to say that

(a - b) - c = a - (b + c)

When we say that addition is associative, all we mean is that it makes no difference what order we do additions in. It does make a difference what order we do subtractions in so subtraction is not an associative operation.

Although mathematicians say that they reason from from axioms that have no necessary relation to the physical world, the fact is that the axioms of arithmetic are massively confirmed by physical experiment. They are not just arbitrary rules for arranging symbols; it is just that pure mathematicians like to think about them that way and ignore the fact that they have physical correlations.

 

[Saumyojit]

Why do you say the 1st case is correct when it obviously is not?
Actually I was saying from the fact that 1st case is right in showing "Sub is non asso"

 

[Saumyojit]

In Post no30
Jeff said that parentheses means grouping symbols specifying order of operation but it is subject to change .

So if "parentheses means grouping symbols specificyiing order of operation" is not a mathematical fact and for
associative as I know the operation is called associative even when you aren't at the moment using any parentheses. Associativity is a property of the operation, not just of an expression.

But if parentheses meant something else then can associative property still be a "mathematical fact" ??

(2+3)+5=2+(3+5)

Here we can show exactly how associativity is working .
But if the meaning of parenthesis is changed Only, (+ will be a associative operation though) I cannot give this eg to show what associative actually means(i.e by changing the Parentheses) to make someone understand

 

[JeffM]

In Post no30
Jeff said that parentheses means grouping symbols specifying order of operation but it is subject to change .

So if "parentheses means grouping symbols specificyiing order of operation" is not a mathematical fact and for
associative as I know the operation is called associative even when you aren't at the moment using any parentheses. Associativity is a property of the operation, not just of an expression.

But if parentheses meant something else then can associative property still be a "mathematical fact" ??

(2+3)+5=2+(3+5)

Here we can show exactly how associativity is working .
But if the meaning of parenthesis is changed Only, (+ will be a associative operation though) I cannot give this eg to show what associative actually means(i.e by changing the Parentheses) to make someone understand

My point was that mathematical notation is, like every language, a super-individual agreement on how sounds or symbols are to be construed. Mathematical results and ideas are not dependent on the notation or language used to express them.

Addition would be associative, and subtraction would not, whether or not we used parentheses as symbols to specify order of operations. The idea is independent of the means used to express it.

 

[Saumyojit]

Addition would be associative, and subtraction would not, whether or not we used parentheses as symbols to specify order of operations. The idea is independent of the means used to express it

Ok addition will be still be a associative operator . I got it the idea is independent of the the means used to express it.
in this expression 2+3 =5 ; addition is a associative operation .
Now given that (parentheses means something else)a expression containing 3 operands
(2+3)+5=2+(3+5)-->I was saying that I cannot show that rearranging parentheses like this would not affect the outcome of the expression; coz the parentheses means something else otherwise i can demonstrate to someone that rearranging parentheses like shown above does not affect the result as + is asso. This was my point.

My point was that mathematical notation is, like every language, a super-individual agreement on how sounds or symbols are to be construed.

language? Means English u are telling.
Sounds ??
I got it that you are referring to Parentheses as "mathematical notation" right?

Mathematical results and ideas are not dependent on the notation or language used to express them

Mathematical ideas which means here "Associativity" right?
Mathematical results means ... what ? in this context
What do you mean by the above quoted para?
please use simple English

 

[Saumyojit]

I edited a bit

 

[JeffM]

Mathematical notation is a language, just like English, Hindustani, Bengali, or Tamil. It is a way to communicate among people. The only differences of any import are that (1) mathematical notation uses symbols rather than words (sounds) and so can be understood by people with different native languages, and (2) mathematical notation contains many fewer ambiguities than natural languages. An important part of what we learn when studying mathematics is this international language.

But every language is simply a set of ARBITRARY agreements about what certain combinations of symbols or sounds mean. To communicate with others, you cannot change those meanings individually. The group that uses those symbols or sounds must agree on any changes. There is nothing magical about parentheses except that they are, under the current agreement, grouping symbols that specify which operations are to be performed first.

But the associativity of the addition of numbers is a property of the operation. Adding 3 and 5 and then adding 8 and 7 results in 15, and adding 5 and 7 and then adding 12 + 3 ALSO results in 15. The results are the same no matter how we order the operations of addition. It grows out of what we mean by numbers and addition. It is not dependent on the notation that we use to explain the idea.

 

[Saumyojit]

The results are the same no matter how we order the operations of addition.

I know.
I was saying that if the expression is like this 2+3+4 I can say what ever order in which I will evaluate this; I will get same answer but if I wanted to show someone who is new to maths that what did I mean by "whatever order" then if parentheses means something else I cannot show them the below format to explain the meaning of "whatever order of evaluation"

(2+3)+5=2+(3+5)
representing two orders of evaluation

I know that answer is same but that was not my point.

 

[HallsofIvy]

Ok addition will be still be a associative operator . I got it the idea is independent of the the means used to express it.
in this expression 2+3 =5 ; addition is a associative operation .
Now given that (parentheses means something else)a expression containing 3 operands
(2+3)+5=2+(3+5)-->I was saying that I cannot show that rearranging parentheses like this would not affect the outcome of the expression; coz the parentheses means something else otherwise i can demonstrate to someone that rearranging parentheses like shown above does not affect the result as + is asso. This was my point.





language? Means English u are telling.
Sounds ??
I got it that you are referring to Parentheses as "mathematical notation" right?


Mathematical ideas which means here "Associativity" right?
Mathematical results means ... what ? in this context
What do you mean by the above quoted para?
please use simple English

I wish you would use simple English! I have no idea what "Means English u are telling." or "Mathematical ideas which means here "Associativity" right?" could mean!

Addition is a "binary" operation. It is defined as adding two numbers at a time. But we have no problem adding three or more numbers because of "associativity". If I want to add, say, 3, 5, and 7 I have to decide which two to add first. "Associativity" tells me it doesn't matter. 3+ 5= 8 and then 8+ 7= 15. Or 5+ 7= 12 and then 3+ 12= 15.

(I could also add 3+ 7= 10 and then 10+ 5=15 but changing the order, from 3+ 5+ 7 to 3+ 7+ 5, is a different property, "commutativity".)

 

[JeffM]

I know.
I was saying that if the expression is like this 2+3+4 I can say what ever order in which I will evaluate this; I will get same answer but if I wanted to show someone who is new to maths that what did I mean by "whatever order" then if parentheses means something else I cannot show them the below format to explain the meaning of "whatever order of evaluation"

(2+3)+5=2+(3+5)
representing two orders of evaluation

I know that answer is same but that was not my point.

Yes, under the current notation, where parentheses are grouping symbols, we can use them to explain the idea of associativity. If they meant something else, we could not use them for that purpose.

But I think you are letting the tail wag the dog. What are truly important are the idea of associativity and the fact that addition of numbers is associative rather than the symbolism used. I’d explain the idea and demonstrate the idea’s applicability to ordinary numbers long before trying to explain the symbolism used to express all that. Math ultimately is about ideas and facts, not symbols.