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Closure

Algebra 2

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. A subset of the real numbers is closed under addition if, for any two numbers, a and b, that are members of the subset, the number is also a member of the subset. Tell whether each of the following subsets of the real numbers is closed under addition. If it is not, give an example that shows it is not. a. The set of whole numbers b. The set of negative integers c. The set of irrational numbers d. The set of rational numbers 2. For each of the sets in Problem 1, tell whether the set is closed under multiplication. If it is not, give an example that shows it is not. 3. a. For each of the following pairs of rational numbers, and find the rational number and write the 3 rational numbers in increasing order. (i) (ii) (iii) (iv)

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Closure Property of Addition Sum (or difference) of 2 real numbers equals a real number Additive Identity a + 0 = a Additive Inverse a + (-a) = 0 Associative of Addition (a + b) + c = a + (b + c) Commutative of Addition a + b = b + a Definition of Subtraction a - b = a + (-b) Closure Property of Multiplication Product (or quotient if denominator 0) of 2 reals equals a real number Multiplicative Identity a * 1 = a Multiplicative Inverse a * (1/a) = 1 ? ? (a 0) (Multiplication times 0) a * 0 = 0 Associative of Multiplication (a * b) * c = a * (b * c) Commutative of Multiplication a * b = b * a Distributive Law a(b + c) = ab + ac Definition of Division a / b = a(1/b)
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