In mathematics, if x, y, z be any three variables involving the addition operation and then satisfy the following condition,
x + (y + z) = (x + y) + z
This kind of property is called associative property sum or addition. In associative property sum, the sum of algebraic expression provides the same result in changing the order of brackets in this expression. Such as
x + y + z = (x + y) + z = x + (y + z)
Examples for associative property sum:
Example 1 on associative property sum:
Prove the associative property sum of the given expression.
(a) 4 + 6 + 7
Solution:
(a) Let x, y, and z be 4, 6 and 7 respectively.
Therefore, the given expression satisfy the following condition,
x + y + z = x + (y + z) = (x + y) + z
Here,
Prove the following expression
4 + (6 + 7) = (4 + 6) + 7
L.H.S = 4 + (6 + 7) (first perform the addition operation within the bracket from left to right,
because brackets are higher priority of the order of operations)
= 4 + (13) (second perform the addition operation outside of the expression from left to right)
= 17 ------ (1)
R.H.S = (4 + 6) + 7
= (10) + 7
= 17 --------- (2)
From equation (1) and (2) we get,
L.H.S = R.H.S
i.e., 4 + (6 + 7) = (4 + 6) + 7
Hence, the given expressions satisfy the associative property with sum of numbers.
Example 2 on associative property sum:
(a) 23 + 67 + 34, prove the associative property of sum or addition.
Solution:
(a) Let m, n, and l be 23, 67 and 34 respectively.
Therefore, the given expression satisfy the following condition,
m + n + l = m + (n + l) = (m + n) + l
Here,
Prove the following expression
23 + 67 + 34 = 23 + (67 + 34) = (23 + 67) + 34
23 + 67 + 34 = 124 ---------------- (1)
We consider the remaining expression, such as
23 + (67 + 34) = (23 + 67) + 34
L.H.S = 23 + (67 + 34)
= 23 + (101)
= 124 ------ (2)
R.H.S = (23 + 67) + 34
= (90) + 34
= 124 --------- (2)
from equation (1) , (2) and (3) we get,
L.H.S = R.H.S = 124
i.e., 4 + (6 + 7) = (4 + 6) + 7 = 4 + 6 + 7
Hence, the given expression of sum is satisfied the associative property.
