The associative rule of addition states that addition can be performed in any order without affecting the resulting number. For instance, 4 + 3 is the same as 3 + 4. Likewise, adding large sequences of numbers such as 4, 83, 222, 45 and 50 can be done in any order, always yielding the same result. This rule is often displayed as a + (b + c) = (a + b) + c.
The associative rule of multiplication is much like that of addition. It states that when the only operation involved is multiplication, this operation can be performed in any sequence. Thus, 3 times 30 is 90, which is also the result of 30 times 3. In a similar manner to that of the associative rule of addition, the associative rule of multiplication is written as a(bc) = (ab)c.
In set theory, the operation "or" allows a given element to belong to one of two sets. For example "A or B" means an element can exist either in A or B, as well as in both A and B simultaneously. When there are more than two sets being considered, set theorists apply the associative rule of "or," which states that you can look at multiple "or" statements in any order you wish. For example, if we are trying to decide what sets element "e" can exist in and we know that "e" can exist in (A or B) or (C), using the associative rule, we know that this statement is equivalent to "A or (B or C)."
Set theory also makes heavy use of the "and" operation, which states that an element must be in both sets of interest. For example, the statement "e is in A and B" means that "e" simultaneously exists in A and B. Again, when dealing with more than two sets, the associative property comes into play. The associative property for the "and" operation is commonly written as (A and B) and C <=> A and (B and C).