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Midterm I Handout

# Midterm I Handout - proof you just need to use it You...

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MATH 74 MIDTERM 1 SHEET + and · are binary operations on R , and (A1) x + ( y + z ) = ( x + y ) + z for all x , y , and z in R (A2) x + y = y + x for all x and y in R (A3) 0 + x = x for all x in R (M1) ( x · y ) · z = x · ( y · z ) for all x , y , and z in R (M2) x · y = y · x for all x and y in R (M3) 1 · x = x for all x in R (D) ( x + y ) · z = ( x · z ) + ( y · z ) for all x , y , and z in R Fact 1. + is also a binary operation on any of: N , Z , Q , C Fact 2. The usual notion of subtraction, - , is a binary operation on any of Z , Q , R , C Fact 3 (Sequence equality theorem) . if A n and B n are sequences of real numbers (defined for n N ) satisfying the two conditions A 1 = B 1 A n +1 - A n = B n +1 - B n for all n N then A n = B n for all n N . Fact 4 (Induction principle) . Suppose that one has for every n N a statement P ( n ) . If the following two conditions are met: P (1) is true Whenever k N has the property that P ( k ) is true, then P ( k + 1) is also true then P ( n ) is true for all n N . Guidelines for proof techniques Proofs by contradiction. You may assume that I am familiar with the logic behind proof by contradiction: you do not need to “explain” the technique in your
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Unformatted text preview: proof, you just need to use it. You should, however • Begin your proof by indicating you will be writing a proof by contradiction, • Make clear what the source of the contradiction is (e.g. “but 1 6 = 0, a contradiction” or “this shows y < 0, but we also know that y ≥ 0, a contradiction”) Proofs by induction. You may assume I understand the induction principle (you do not need to explain the logic of “why” it works in your proof). You should, however, begin your proof by saying that you will use the induction principle, and you might like to explicitly identify what P ( n ) is. (I don’t require this last point, but it sometimes is helpful in making the structure of the proof clearer to the reader.)...
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