klarfeld-sec-4-3-4-4 - Jennifer Klarfeld Section 4.3-4.4...

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Jennifer’Klarfeld,’Section’4.3 -4.4,’Edition’8’ 1 SECTION’4.3 ’ 5.’Prove’that’if’ ࠵:± ⟶ ² ’ is’onto’and’ ³:² ⟶ ´ ’ is’onto,’then’ ³ ∘ ࠵:± ⟶ ´ ’ is’onto. ’ By’Theorem ’ 4.2.1,’ µ¶· ³ ∘ ࠵ = µ¶· = ±. For’all’ ¸ ∈ ´ ,’because’ ³ ’ is’onto,’there’exists’ ¹ ∈ ² ’ such’that’ ³ ¹ = ¸ .’ Since’ ’ is’onto,’for’that’ ¹ ∈ ² ,’there’ exists’ º ∈ ± ’ such’that’ º = ¹. Therefore,’for’all’ ¸ ∈ ´ ,’there’exists’ º ∈ ± ’ such’that’ ³ ∘ ࠵ º = ³ º = ³ ¹ = ¸. Thus,’ ³ ∘ ࠵:± ⟶ ´ ’ is’onto. ’
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Jennifer’Klarfeld,’Section’4.3 -4.4,’Edition’8’ 2 6.’Prove’that’if’ ࠵:± ⟶ ² ,’ ³:² ⟶ ´ ,’and’ ³ ∘ ࠵:± ⟶ µ3µ ´ ,’then’ ࠵:± ⟶ µ3µ ² .’ Suppose’ = ࠵(·) ’ for’some’ ¶,· ∈ ± .’ Then’ ³ ∘ ࠵ = ³ = ³ · = ³ ∘ ࠵ · . Since’ ³ ∘ ࠵ ’ is’one -to-one,’ ³ ∘ ࠵ = ³ ∘ ࠵ · ’ implies’ ¶ = ·. Thus,’ ’ is’one -to-one.’
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Jennifer’Klarfeld,’Section’4.3 -4.4,’Edition’8’ 3 7.’Prove’that’if’ ࠵:± ⟶ ² ’ is’one -to-one,’then’every’restric tion’of’ ’ is’one -to-one.’ Let’ ’ be’one -to-one’and’ ³ ⊆ ±. Consider’ ࠵| ³ = ´,µ ∶ ࠵ ´ = µ ¶·¸ ´ ∈ ³ . Suppose’that’ ࠵| ³ = ࠵| ³ ¹ ࠵º» ¼º½¾ ¶,¹ ∈ ³. Then’ (¶,¿) ∈ ࠵| ³ ’ and’ (¹,¿) ∈ ࠵| ³ ’ with’ ¿ = ࠵| ³ = ࠵| ³ ¹ . Since’ ࠵| ³ ⊆ ࠵ ,’ (¶,¿) ∈ ࠵ ’ and’ (¹,¿) ∈ ࠵ .’ Thus,’because’ ’ is’one -to-one,’ = ¿ = ࠵(¹) ’ implies’ ¶ = ¹. Therefore,’ ࠵| ³ ’ is’one -to-one.’ We’conclude’that’every’restriction’of’ ’ is’one -to-one.’
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Jennifer’Klarfeld,’Section’4.3 -4.4,’Edition’8’ 4 8.’Prove’part’(a)’of’Theorem’4.3.6:’Le t’ ࠵:± ⟶ ² ’ and’ ³:´ ⟶ µ ’ be’functions.’If’ ± ’ and’ ´ ’ are’ disjoint’sets,’ ’ is’onto’ ² ,’and’ ³ ’ is’onto’ µ ,’then’ ࠵ ∪ ³ ∶ ± ∪ ´ ⟶ ² ∪ µ ’ is’onto’ ² ∪ µ. Suppose’ ± ∩ ´ = ∅ ,’ ’ is’onto’ ² ,’and’ ³ ’ is’onto’ µ .’ By’Theorem’4.2.5,’ ࠵ ∪ ³
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