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Unformatted text preview: 1.3 The rational numbers 23 3. ∀ x ∈ G, ∃ ! y ∈ G such that x + y = y + x = 0. Write y = x . Written multiplicatively, ( G, × , 1) must satisfy 1. x,y ∈ G = ⇒ x × y is a welldefined element of G . 2. ∃ !1 such that x × y = y × x = x, ∀ x ∈ G . 3. ∀ x ∈ G, ∃ ! y ∈ G such that x × y = y × x = 1. Write y = 1 x . Theorem 1.3.2. Z , Q , R , C are groups under addition. N , N are not. Theorem 1.3.3. Let Q × = { x ∈ Q . . . x 6 = 0 } . Then Q × is a group under multiplication. So are R × and C × . Z × is not. Definition 1.3.4. A set which is a group under addition, and whose nonzero elements form a group under multiplication is called a field if the two operations behave nicely together: a × ( b + c ) = ( a × b ) + ( a × c ) (Distributive law) and the operations + , × are commutative. Theorem 1.3.5. Q , R , C are fields. GL n = { invertible n × n matrices } is not....
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 Fall '11
 Wong
 Equivalence relation, Binary relation, Total order

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