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In mathematics, the associative law is a property of some binary operations. It states that the order in which the operations are performed does not matter as long as the order of the operands is not changed. In other words, the associative law allows us to rearrange the parentheses in an expression without affecting the result.

For example, consider the expression (4 + 5) + 6. Using the associative law, we can rearrange the parentheses to get the equivalent expression 4 + (5 + 6). Both expressions will produce the same result, which is 15.

The associative law is often used to simplify complex expressions by grouping similar operations together. For example, the expression 4 + 5 + 6 + 7 can be simplified using the associative law as follows: (4 + 5) + (6 + 7) = 9 + 13 = 22.

The associative law is applicable to many operations, including addition, multiplication, and composition of functions. It is not applicable to all operations, however. For example, the operation of subtraction is not associative.

Here are some more examples of the associative law:

Example 1: (4 + 5) + 6 = 4 + (5 + 6)

Example 2: (4 * 5) * 6 = 4 * (5 * 6)

Example 3: (f ? g) ? h = f ? (g ? h)

Example 4: [(4 + 5) + 6] * 7 = 4 + [5 + (6 * 7)]

Example 5: 4 * (5 + 6) * 7 ? (4 * 5) + (6 * 7)

Quiz:

1. Is the associative law applicable to the operation of subtraction?
2. Can the associative law be used to simplify the expression 4 + 5 + 6 + 7?
3. Is the expression (4 + 5) + 6 equivalent to 4 + (5 + 6)?
4. Is the expression [(4 + 5) + 6] * 7 equivalent to 4 + [5 + (6 * 7)]?
5. Can the associative law be used to rearrange the parentheses in the expression (4 * 5) * 6 * 7?
6. Is the operation of composition of functions associative?
7. Can the associative law be used to simplify the expression (4 * 5) + (6 * 7)?
8. Is the expression (4 + 5) * 6 equivalent to 4 * (5 + 6)?
9. Is the expression (f ? g) ? h equivalent to f ? (g ? h)?
10. Can the associative law be used to rearrange the parentheses in the expression 4 + 5 + 6?

Answers:

1. No
2. Yes
3. Yes
4. Yes
5. Yes
6. Yes
7. No
8. No
9. Yes
10. Yes