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Quiz3_sol - ′ c ′ b ′ d ′ a b ∼ a ′ b ′...

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QUIZ 3 SOLUTIONS WED., OCT. 13 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points Score 1 1 2 2 3 2 Total 5 1
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QUIZ 3 SOLUTIONS WED., OCT. 13 NAME: Problem 1: (1 point) What is the definition of an onto or surjective function? Let f : A B be a function. Then if for all b B there exists an a A such that f ( a ) = b , we call f onto or surjective. Problem 2: (2 points) Let { ( a,b ) } , { ( c,d ) } be equivalence classes in Q . Recall the definition of the equiv- alence relation ( p,q ) ( p ,q ) ⇐⇒ pq = qp . Define multiplication in Q as { ( a,b ) } · { ( c,d ) } = { ( ac,bd ) } . Prove that multiplication is a well defined operation. Proof: We must show that for ( a,b ) ( a ,b ) and ( c,d ) ( c ,d ) then ( ac,bd ) ( a
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Unformatted text preview: ′ , c ′ , b ′ d ′ ). ( a, b ) ∼ ( a ′ , b ′ ) ⇐⇒ ab ′ = a ′ b, (1) ( c, d ) ∼ ( c ′ , d ′ ) ⇐⇒ cd ′ = c ′ d. (2) Multiplying the frst by cd ′ and substituting in the second gives ab ′ cd ′ = a ′ bcd ′ = a ′ bc ′ d. Q. E. D. Problem 3: (2 points) Label the Following true or False (a) (0.5 points) T Q is an ordered feld. (b) (0.5 points) ± A feld need not be an integral domain. (c) (0.5 points) ± Let X be a nonempty set and ℘ ( X ) be the power set oF X . Then ℘ ( X ) with △ as ‘+’ and ∩ as ‘ · ’ is a feld. (d) (0.5 points) ± A subset oF A × B such that each element oF A occurs exactly once as a frst coordinate is called an injective Function. 2...
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