# quiz3bANS - ∅ so it is indeed an element b{∅ ∈{∅...

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Math 240, Quiz 3 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1) Prove that if x is rational and x n = 0, then 1 /x is rational. If x is a nonzero rational, then we can write x as a b with a, b Z , and a n = 0 , b n = 0. Then 1 x = 1 a b = b a . This is clearly a rational number. Note: several people wrote x R . This means x is a real number, and not necessarily rational. It should be written x Q . 2) Determine whether these statements are true or false (and justify your conclusion). a) ∅ ∈ {∅ , {∅}} This is true. The Frst element of the set on the right is
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Unformatted text preview: ∅ , so it is indeed an element. b) {∅} ∈ {∅} This is not true. The sets are in fact equal, and it is never true that a set is an element of itself. Remember, this is di±erent from the true statment that S ⊆ S . c) {{∅}} ⊆ {∅ , {∅}} This is true. The only element of the right hand side is {∅} . This is the second element in the right hand side. Thus every element of the left hand side is an element of the right hand side. 1...
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