Unformatted text preview: ( n ) = n +1 is injective but not surjective. Now deFne σ ( n ) = ± n1 if n ± = 0 if n = 0. Then τ is clearly surjective, but not injective as τ (0) = τ (1) = 0. 9. Let A = { a } , B = { b, c } and C = { d } . DeFne σ : A→ B by the rule σ ( a ) = b and τ : B→ C by the rule τ ( b ) = τ ( c ) = d . Then the composition τ ◦ σ : A→ C sends a to d . Thus the composition is bijective and yet σ is not surjective and τ is not injective. 2. Suppose that A n are countable sets, n ∈ N . We want to prove that A = ² n ∈ N A n is countable. We might as well assume that this is a disjoint union. Pick injections f n A n→ N . DeFne a map f : A→ N × N , by the rule f ( a ) = ( n, f n ( a )) where a ∈ A n . It is clear that f is an injection, and the result is an easy application of Schr¨oderBernstein. 1...
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This note was uploaded on 07/15/2008 for the course MATH 111 taught by Professor Long during the Spring '08 term at UCSB.
 Spring '08
 LONG
 Algebra

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