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# quiz3aANS - the set that has exactly one element that...

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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 the product of two rational numbers is rational. Let x, y be rational (a side note: if x, y are rational, that means x, y Q , not, as several people wrote, x, y R ). Then we can write x = a b and y = c d , with a, b, c, d Z , and b negationslash = 0 , d negationslash = 0. Then xy = a b · c d = ac bd , with bd negationslash = 0, and ac, bd Z . Thus xy a rational number. 2) Determine whether these statements are true or false (and justify your conclusion). a) ∅ ∈ {∅} This is true. Many people were confused. Now,
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Unformatted text preview: the set that has exactly one element, that element being the empty set. ∅ has no elements. {∅} has exactly one element. Since ∅ is in fact a member of {∅} , this statement is true. b) {∅} ∈ {{∅}} This statement is also true. Note that neither of these sets is the empty set. c) {{∅}} ⊆ {{∅} , {∅}} This is true. In fact, the sets are equal (note that the right hand side has {∅} listed twice. Since we don’t care about repetition, we could simply write the right hand side as {{∅}} ). 1...
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