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# Associative Property

Associative property is a basic property of the binary operations which is used in mathematics. This property says that if we perform addition or multiplication operation on any Set of Numbers then both the operations will be same without considering the grouping of the numbers i.e. it does not matter how numbers are grouped. This property basically includes more than two numbers to perform operations. To indicate each unit of a given set of numbers; it uses parenthesis. Each parenthesis is used for grouping the numbers. Grouping of numbers into parentheses is the main part of ordering of operations.
It rearranges the parentheses in such a way that the expression does not change. We can simply define associative property by a binary operation say * multiplication on a given set of numbers say N is called associative if and only if it satisfies the associative law which says:
(x * y) * z = x * (y * z)
Where * is the binary operation and x, y, z are the numbers of a given set or x, y, z € N. In these type of expressions the order of evaluation does not matter for solving them.
We can also write the above expression as:
(xy)z = xyz
Means it is simple xyz that indicates that even if we omit the parentheses it will not change the actual computation of the expression. Also we must remember that we can only change the order of operation. We are not allowed to change the order of operands within the expression. The sequence of operands must remain same.
Similarly in case of binary operation + or addition this can be given by a law of associative property as:
(x + y) + z = x + (y + z)
Here we change the order of parentheses that means the order of operation and the operands are unchanged; still it will give the same result.
In Math associative property can be written in the form of functional notation as:
F (F(x, y), z) = F(x, F(y, z)); here F denotes the function.
The associative property of math is not similar to Commutative Property; there is a slight difference between these two properties. The commutative property changes the order of operands in an expression while associative property changes the order of operation. It can also be performed on n- ary operations where number of operands or numbers are more than three.

## If we have three numbers say x = 4, y = 2 and z = 5 then show that Associative Property holds true for addition?

According to Associative Property we know that (x + y) + z = x + (y +z)
So we observe by putting the above given values in the given equation that:
L.H.S. = (4 + 2) + 5        R.H.S. = 4 + (2 + 5),
= 6 + 5                              = 4 + 7,
= 11                                  = 11,
So L.H.S = R.H.S. (So Associative Property holds true for addition.)

## If we have three positive natural numbers say x = 4, y = 2 and z = 5 then check that Associative Property holds true for subtraction or not?

According to Associative Property we know that (x - y) - z = x - (y -z)
So we observe by putting the above given values in the given equation that:
L.H.S. = (4 - 2) - 5        R.H.S. = 4 - (2 - 5),
= 2 – 5              and       = 4 – (-3),
= -3                    and       = 4 + 3 = 7,
So we get L.H.S <> R.H.S.
So we conclude that associative property does not hold true for subtraction.

## If we have three fraction numbers say x = 4/5, y = 2/5 and z = 3/5 then show that Associative Property holds true for addition?

According to Associative Property we know that (x + y) + z = x + (y +z)
SO we observe by putting the above given values in the given equation that:
L.H.S.= (4/5 + 2/5) + 3/5         R.H.S.= 4/5 + (2/5 + 3/5),
= (4 + 2)/5 + 3/5                          = 4/5 + (2 + 3)/5,
= 6/5   + 3/5                                 = 4/5 + 5/5,
= (6 + 3)/ 5                                   = (4 + 5)/5,
= 9/5                                             = 9/5,
So associative property holds true for the Fractions addition

## If we have three fraction numbers say a = 4/7, b = 2/7 and c = 6/7 then show that Associative Property holds true for subtraction of fraction numbers or not?

According to Associative Property we know that (a - b) - c = a - (b - c)
SO we observe by putting the above given values in the given equation that:
L.H.S. = (4/5 - 2/5) - 3/5         R.H.S.= 4/5 - (2/5 - 3/5),
= (4 - 2)/5 - 3/5                           = 4/5 - (2 - 3)/5,
= 2/5 – 3/5                                  = 4/5 – (-1/5),
= -1/5                                           = 4/5 + 1/5,
= 5/5.
Thus we observe that the two results are different. So the associative property does not hold true for the subtraction of the fraction Numbers.

## If we have three numbers say x = 4, y = 2 and z = 5 then show that Associative Property holds true for multiplication?

According to Associative Property we know that (x * y) * z = x * (y * z)
SO we observe by putting the above given values in the given equation that:
L.H.S. = (4 * 2) * 5        R.H.S. = 4 * (2 * 5),
= 8 * 5                          = 4 * 10,
= 40                              = 40
So the property of association holds true for multiplication of natural Numbers.

## If we have three numbers say x = 2, y = 3 and z = 4 then show that Associative Property holds true for division?

According to Associative Property we know that (x / y) / z = x / (y /z).
So we observe by putting the above given values in the given equation that:
L.H.S. = (2/3) / 5        R.H.S. = 4 / (2 / 5),
In the above case we observe that the LHS <> RHS so the property of division does not hold true for the division operation.

## Does there exist any values of the three numbers x, y, and z such that the associative property holds true for division?

If we take the value of x= 1, y = 1 and z = 1, then we can check on putting these values in
(x / y) / z = x / (y /z), we get
LHS = (1/1) /1 and RHS = 1/ (1/1),
So LHS = 1 and RHS = 1 ,
So the property of division holds true. This is only possible when all the three Numbers have the value 1, 1 and 1.

## Does there exist any values of the three numbers x, y, and z such that the associative property holds true for subtraction. If yes, find any one set of such three numbers?

If we take the value of x= 1, y = 0 and z = 0, then we can check on putting these values in
(x - y) - z = x - (y - z), we get,
LHS = (1- 0) - 0 and RHS = 1 - (0 – 0),
So LHS = 1 and RHS = 1. So the property of subtraction holds true. This is also possible when all the three Numbers have the value 0.

## What do you mean by Associative property of Addition of integers?

According to Associative Property of Addition we Mean that if a, b and c are three integers then we say,
(a + b) + c = a + (b + c).

## What do you mean by Associative property of multiplication of integers?

According to Associative Property of Addition we Mean that if a, b and c is any three integers then we say
(a * b) * c = a * (b * c).

## If we have three numbers say x = 4, y = 2 and z = 5 then show that associative property holds true for addition?

According to Associative Property we know that:
(x + y) + z = x + (y + z)
So, we observe by putting the above given values in the given equation that:
L.H.S. = (4 + 2) + 5             R.H.S. = 4 + (2 + 5),
= 6 + 5                                    = 4 + 7,
= 11                                        = 11,
So L.H.S = R.H.S.
Thus, we can say that  associative property holds true for addition.

## If we have three positive natural numbers say x = 4, y = 2 and z = 5 then check that associative property holds true for subtraction or not?

According to Associative Property we know that (x - y) - z = x - (y -z)
So, we observe by putting the above given values in the given equation that:
L.H.S. = (4 - 2) - 5        R.H.S. = 4 - (2 - 5),
= 2 – 5              and        = 4 – (-3),
= -3                 and           = 4 + 3 = 7,
So we get L.H.S <> R.H.S. So we conclude that associative property does not hold true for subtraction.

## If we have three fraction numbers say x = 4/5, y = 2/5 and z = 3/5 then show that associative property holds true for addition?

According to Associative Property we know that (x + y) + z = x + (y +z)
SO we observe by putting the above given values in the given equation that:
L.H.S.= (4/5 + 2/5) + 3/5     R.H.S.= 4/5 + (2/5 + 3/5),
= (4 + 2)/5 + 3/5                    = 4/5 + (2 + 3)/5,
= 6/5   + 3/5                           = 4/5 + 5/5,
= (6 + 3)/ 5                             = (4 + 5)/5,
= 9/5                                       = 9/5.
So, associative property holds true for the Fractions addition.

## If we have three fraction numbers say a = 4/7, b = 2/7 and c = 6/7 then show that associative property holds true for subtraction of fraction numbers or not?

According to Associative Property we know that (a - b) - c = a - (b - c)
So, we observe by putting the above given values in the given equation that:
L.H.S. = (4/5 - 2/5) - 3/5     R.H.S.= 4/5 - (2/5 - 3/5),
= (4 - 2)/5 - 3/5                      = 4/5 - (2 - 3)/5,
= 2/5 – 3/5                            = 4/5 – (-1/5),
= -1/5                                     = 4/5 + 1/5,
= 5/5.
Thus we observe that the two results are different. So the associative property does not hold true for the subtraction of the fraction Numbers.

## If we have three numbers say x = 4, y = 2 and z = 5 then show that associative property holds true for multiplication?

According to Associative Property we know that (x * y) * z = x * (y * z)
So, we observe by putting the above given values in the given equation that :
L.H.S. = (4 * 2) * 5        R.H.S. = 4 * (2 * 5),
= 8 * 5                              = 4 * 10,
= 40                                  = 40.
So the property of association holds true for multiplication of natural Numbers.

## If we have three numbers say x = 2, y = 3 and z = 4 then show that associative property holds true for division?

According to Associative Property we know that (x / y) / z = x / (y /z).
SO we observe by putting the above given values in the given equation that:
L.H.S. = (2/3) / 5        R.H.S. = 4 / (2 / 5),
In the above case we observe that the LHS <> RHS.
So. the property of division does not hold true for the division operation.

## Does there exist any values of the three numbers x, y, and z such that the associative property holds true for division?

If we take the value of x= 1, y = 1 and z = 1, then we can check on putting these values in
(x / y) / z = x / (y /z), we get
LHS = (1/1) /1 and RHS = 1/ (1/1),
So LHS = 1 and RHS = 1 ,
So the property of division holds true. This is only possible when all the three Numbers have the value 1, 1 and 1.

## Does there exist any values of the three numbers x, y, and z such that the associative property holds true for subtraction. If yes, find any one set of such three numbers?

If we take the value of x= 1, y = 0 and z = 0, then we can check on putting these values in
(x - y) - z = x - (y - z), we get,
LHS = (1- 0) - 0 and RHS = 1 - (0 – 0),
So LHS = 1 and RHS = 1.
So the property of subtraction holds true. This is also possible when all the three Numbers have the value 0.

## What do you mean by Associative property of Addition of integers?

According to Associative Property of Addition we Mean that if a, b and c are three integers then we say,
(a + b) + c = a + (b + c).

## What do you mean by Associative property of multiplication of integers?

According to Associative Property of Addition we Mean that if a, b and c is any three integers then we say:
(a * b) * c = a * (b * c).