Review1-2011

# Review1-2011 - brieﬂy 8 Let x ∈ Z Prove the following 2...

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MATH 127: Exam 1 Review Monday, February 7, 2011 1. Let x,y,z R . Show that x 2 + y 2 + z 2 2 x + 4 y + 6 z - 14. 2. Prove that if x,y N and x 2 - y 2 = 3 then x = 2 and y = 1. 3. Find, with proof, all pairs of natural numbers x and y satisfying x 2 - y 2 = 15 4. Use double containment to show the set equality A - B = A B c 5. Prove or Disprove: for all sets A , B and C , ( A B ) C = A ( B C ) 6. Translate the following sentences into symbolic form (specifying the meaning of your predi- cates). In each case the universe is written in parentheses. One of my friends does not have a car. (all people) At least on British snake is poisonous. (all snakes) All elephants drink water. (all animals) 7. Let A = { 1 , 2 , 3 } , B = { 4 , 5 , 6 } and C = { 1 , 5 , 7 } . Consider the following statements: (i) ( x A )( y B )( z C )( x + y = z ) (ii) ( x A )( y B )( z C )( x + y z ) (iii) ( x A )( y B )( z C )( x 2 + y 2 = z 2 ) (iv) ( x A )( y B )( z C )(6 x - y < z ) (a) Negate each of these sentences symbolically, writing your answer in positive form (i.e. without using ¬ ). (b) Which of the original sentences were true and which were false? Explain your answer

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Unformatted text preview: brieﬂy. 8. Let x ∈ Z . Prove the following: 2 x 2 + x + 1 is even iﬀ x is odd. 9. Let m,n ∈ Z . Show that there doesn’t exist an integer k such that (3 m +2)(3 n +2) = 3 k +2. 10. Let x ∈ R . Prove the following: If | x-1 | < 1 then | x 2-4 x + 3 | < 3 11. Let h a n | n ∈ N i be the sequence deﬁned recursively as follows: a 1 = 1 a n +1 = a n + 3 n ( n + 1) for n ∈ N Prove that a n = n 3-n + 1 for all n ∈ N . 12. Show that ∀ n ∈ N the following holds: n X k =1 2 k = 2 n +1-2 13. Recall that the Fibonacci Numbers are deﬁned by f = 0, f 1 = 1 and f n = f n-1 + f n-2 for all n > 1. Show that for every natural number n the following holds: 2 n-1 f + 2 n-2 f 1 + ... + 2 f n-1 = 2 n-f n +2...
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Review1-2011 - brieﬂy 8 Let x ∈ Z Prove the following 2...

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