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Unformatted text preview: MATH 109 Assignment 3 Due: 4/18/11 #7.2 (i) Counterexample: This is the statement that if m ∈ Z + then { n ∈ Z + : m ≤ n } = Z + . When m = 2, we have 1 / ∈ { n ∈ Z + : 2 ≤ n } so the set does not equal Z + . (iii) Proof. This is the statement that if m ∈ Z + then the set { n ∈ Z + : m ≤ n } is nonempty. Since m ∈ { n ∈ Z + : m ≤ n } , we see that the statement is true for all integers m ∈ Z + . (v) Proof. This is the statement that if n ∈ Z + then the set { m ∈ Z + : m ≤ n } is nonempty. Since n ∈ { m ∈ Z + : m ≤ n } , we see that the statement is true for all integers n ∈ Z + . #7.4 (iii) Proof. This is the statement that if x ∈ R then the set { y ∈ R : xy = 0 } is nonempty. Since x · 0 = 0, we see that 0 ∈ { y ∈ R : xy = 0 } , and hence the set is nonempty. (iv) Proof. This is the statement that the set { y ∈ R : ∀ x ∈ R , xy = 0 } is nonempty....
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This note was uploaded on 08/02/2011 for the course MATH 109 taught by Professor Knutson during the Spring '06 term at UCSD.
 Spring '06
 Knutson
 Math

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