Question

Consider the following statements in a mathematical context where the domain of all variables is specified as the set of integers. Evaluate the truth of the statements:

a) The domain of all variables is the set of integers.
b) For all integers y, and z, the equation x(yz)=(xy)z holds true.

Examine each statement, discussing the implications of the specified domain and assessing the validity of the mathematical equation based on the given conditions. Provide a thorough analysis of the logical and mathematical principles underlying the statements.

Answer 1

The statement utilises the associative property of multiplication which dictates that the way integers are grouped when multiplying does not affect their product, confirming the equation x(yz) = (xy)z is true for all integers.

Step-by-step explanation:

The subject of the question is a mathematical exploration involving the properties of basic arithmetic operations within the domain of integers. It evaluates the associative property of multiplication in a given statement. The associative property states that the way in which factors are grouped in multiplication does not change the product. Given this property, for all integers y, and z, the equation x(yz) = (xy)z indeed holds true, regardless of the integers chosen for x, y, and z. This statement is always valid in the context of integers, which means that within the domain specified, the equation accurately represents a fundamental rule of arithmetic.