Definition of associative property:
Associative property pertains to number, sets and many other mathematical components. The word associative is derived from the word ‘association’, which means being related to. In math, we define associative property for various operations on numbers or sets etc.
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| Associative Property |
What is associative propery?
All types of numbers, whether natural, whole, rational, irrational and real numbers have associative properties. There are mainly two types of associative property for numbers. They are:
1. Associative property of addition:
Addition associative property definition is as follows: Consider three numbers a, b and c. then, (a+b) + c = a + (b+c). Meaning we associate a with b first then associate the sum with c or we associate a with sum of b and c, either ways, the result would be the same.
For example: (1/2 + 1/3) + ¼ = ½ + (1/3 + ¼) = 13/12
2. Associative property of multiplication:
Numbers obey associative property in multiplication as well. Consider three numbers, p,q and r, then (p x q) x r = p x (q x r). That means whether we take product of p and q and then multiply with r or we multiply p with product of q and r, either ways we would get the same answer.
For example: (1/2 x 1/3) x ¼ = ½ x (1/3 x ¼) = 1/24
Associative property with respect to sets:
Consider three non disjoint sets, A, B and C. Then both the operations of union and intersection on these sets is associative. That means, A U (B U C) = (A U B) U C and
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
Example: Consider the following sets: A = {1,2,3,4,5}, B = {2,4,6,8,10}, C = {3,4,5,6,7}
Then, A U (B U C) = {1,2,3,4,5} U ({2,4,6,8,10} U {3,4,5,6,7}) = {1,2,3,4,5,6,7,8,10}
And (A U B) U C = ({1,2,3,4,5} U {2,4,6,8,10}) U {3,4,5,6,7} = {1,2,3,4,5,6,7,8,10}
So we see that both are equal. Similarly .
A ∩ (B ∩ C) = {1,2,3,4,5} ∩ ({2,4,6,8,10} ∩ {3,4,5,6,7}) = {4,6} and
(A ∩ B) ∩ C = ({1,2,3,4,5} ∩ {2,4,6,8,10}) ∩ {3,4,5,6,7} = {4,6}
