Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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MAT102S - Introduction to Mathematical Proofs - Spring 2010 - UTM Problem Set F - NOT to be submitted. The Term Test, to be written on February 24, from 10:10am to 11:00am at CC1080 will cover Problem Sets A-F, and the material discussed in lectures until February 11, 2010. Work on the following exercises: ± Chapter 3, Exercises 3.3, 3.16, 3.21, 3.56. Practice Questions for the Term Test 1. Prove that j p x ² p y j ³ p j x ² y j for all x;y ´ 0 . 2. Let f : A ! B be a function. Show that f ( C [ D ) = f ( C ) [ f ( D ) for any C;D µ A . 3. Prove or give a counterexample: For any function f : A ! B and D µ A , we have f ( D c ) = f ( D ) c . 4. Which of the following are ±elds? Explain. N ; Z ; Q ; [0 ; 1 ) 5. Show that in any ±eld, the equation x 2 = 1 has at most two solutions. Is it possible to have less that two distinct solutions to the equation x 2 = 1 ? 6. Write the negation of the following statement (without using the symbol : ): P = ( 9 x 2 R )((( 9 y 2 R )( x = (1 ² y ) 2 )) ^ (( 9 z 2 R )( x = ² z 2 ))) Which statement is true, P or : P ? 7. Prove that for any three sets
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Unformatted text preview: A;B;C we have ( A B ) \ ( A C ) \ ( B [ C ) = . 8. Show by induction that 4 n + 24 n 10 is divisible by 18 for every n 2 N . 9. Let P ( x ) be assertion \ x is prime", E ( x ) be \ x is even", and D ( x;y ) be \ x divides y " (i.e., y x is an integer). Consider the following statement: R = ( 8 x 2 Z )( P ( x ) ) (( 9 y 2 Z )( E ( y ) ^ D ( x;y )))) Write the negation of R , and determine which statement is true, R or : R . 10. Show that for every n 2 N , 3 4 n +2 + 1 is divisible by 10 . 11. If P and Q are two statements, nd among the following statements, the negation of P , Q , and one which is equivalent to P , Q . Justify your answer: ( : P ) , Q ; ( : P ) , ( : Q ) ; ( P ) Q ) _ ( Q ) P ) ; P ^ Q 12. Prove that for any three sets A;B;C : A ( B C ) = ( A B ) [ ( A \ C ) ....
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