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Unformatted text preview: Math 131a Midterm 1 Lecture 2 Spring 2009 Name: Instructions: • There are 4 problems. Make sure you are not missing any pages. • Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the book covered by this test (excluding the exercises). • Give complete, convincing, and clear answers (or points will be deducted). • No calculators, books, or notes are allowed. • Answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Question Points Score 1 10 2 10 3 10 4 10 Total: 40 1. (10 points) Suppose that f : C → D is an injective function, and that A,B ⊂ C. Prove that f ( A ∩ B ) = f ( A ) ∩ f ( B ) . (Here f ( A ) = { f ( x ) : x ∈ A } and similarly for B and A ∩ B .) Solution: First, we show f ( A ∩ B ) ⊂ f ( A ) ∩ f ( B ) . Suppose that y ∈ f ( A ∩ B ) . Then by definition, there is an x ∈ A ∩ B with f ( x ) = y. Since x ∈ A, we then have y ∈ f...
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 Spring '09
 WEISBART
 Math, Metric space, lower bound, Limit of a sequence

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