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Analysis Worksheet Properties Review Name______________________ Period______Date____________ I. Closure: A set is closed under a given binary operation iff you can take two elements out of the set, perform the operation on them, and the result still belongs in the set. In general: If {a, b} S, then a * b S and b * a S For example, the set of integers is closed under addition (I, +) because if we add two integers, we always get an integer. The set of integers is not closed under division (I, ) because although {7, 4} I, 74 = 1.75 I. (That is a counterexample.) Which of the following are closed? If they are not, provide a counterexample. 1. (N, +) 2. (W, ) 3. (Q, X) 4. (C, X) 5. (Odd integers, X) 6. (Odd integers, +) II. Associative: The associative property means that (a * b) * c = a * (b * c). Which of the following are associative? If not, give a counterexample. 1. (N, +) 2. (W, ) 3. (Q, X) 4. (R, ) 5. (Even integers, +) 6. (Im, ) III. Identity: The identity property means that there is a special element of your set that will allow you to operate with any other element in your set and not alter the other element. I.e., there exists an e S so that a * e = e * a = a for all a S 1. What is the element for addition?__________ 2. To what sets of numbers does this element belong?____________________ Only these sets have an additive identity. 3. What is the element for multiplication?_________ 4. To what sets of numbers does this element belong?____________________ Only these sets have a multiplicative identity. 5. Do any other operations have an identity?_____If so, which ones?_________ IV. Inverse: A set have inverses if, for all non-identity elements in the set, there's another element in the set so that when you operate on these numbers together, you get the identity element. I.e. For all a S, there exists an a-1 S, so that a * a-1 = a-1 * a = e. 1. What is the additive inverse for 6?______ 2. What sets contain all of their additive inverses?___________________ Only these sets have the additive inverse property. 3. What is the multiplicative inverse for 6?_______ 4. What sets contain all of their mutiplicative inverses?________________ Only these sets have the multiplicative inverse property. V. Commutative: The commutative property states that the order in which you operate on two elements doesn't matter. I.e. a * b = b * a for all {a, b, } S. (Note: The identity and inverse properties must be commutative in order to be identities and inverses!) Which of the following are commutative? If one isn't, provide a counterexample. 1. (N, +) 2. (Q, ) 3. (I, X) 4. (R, ) 5. ( Transcendental, +) 6. (Even integers, ) VI. Distributive: The distributive property requires two operations and states that given these two operations, & and %, that a & ( b % c) = (a & b) % (a & c). 1. Write the distributive property of multiplication over addition. ______________________________ Is this property true?_______ 2. Write the distributive property of multiplication over subtraction. ______________________________ Is this property true?________ 3. Write the distributive property of division over addition. ______________________________ Is this property true?________ 4. Write the distributive property of addition over multiplication. ______________________________ Is this property true?________ VII. Groups: If a set and an operation have the properties of closure, associativity, identity and inverse, then they form a group. Which of the following are groups? If they don't, provide a counterexample. 1. (W, +) 2. (Ir, X) 3. (R, ) 4. (C, ) 5. (Even integers, +) 6. (Odd integers, X) VIII. If a set and an operation form a group and the commutative property also holds, then they form an Abelian group. Name two examples of Abelian groups and one group that is not Abelian. 1. 2. 3. IX. A field is formed by a set and TWO operations if the set with both operations forms an Abelian group and there is a distributive property. Which sets of numbers form a field and which two operations must be used?