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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, 74 = 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?
```
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