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# tut1 - Can you extend this procedure to an arbitrary number...

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CS105L: Discrete Structures I semester, 2005-06 Tutorial Sheet 1: Propositional Logic Instructor: Amitabha Bagchi August 7, 2005 All problems in this sheet are from Bogart, Drysdale and Stein’s book. The numbers in bold in the parentheses indicate the location of the problem in that book. 1. ( 3.1, Prob 1 ) Give truth tables for the following expressions: (a) ( s t ) ( ¬ s t ) ( s ∨ ¬ t ) (b) ( s t ) ( t u ) (c) ( s t u ) ( s ∨ ¬ t u ) 2. ( 3.1, Prob 8 ) Use a truth table to show that ( s t ) ( u v ) is equivalent to ( s u ) ( s v ) ( t u ) ( t v ) . 3. ( 3.1, Prob 9 ) Use DeMorgan’s Law, the distributive law and the previous problem to show that ¬ (( s t ) ( s ∨ ¬ t )) is equivalent to ¬ s . 4. ( 3.1, Prob 13 ) Suppose that for each line of a 2-variable truth table you are told the value in the final column, true or false. (For example, you might be told that the final column contains T, F, T and F in that order.) Explain how to create a logical statement using the symbols s, t, , , and
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Unformatted text preview: Can you extend this procedure to an arbitrary number of variables? 5. ( 3.2, Prob 1 ) ±or what positive integers x is the statement ( x-2) 2 + 1 ≤ 2 true? ±or what integers is it true? ±or what real numbers is it true? If we expand the universe for which we are considering a statement about a variable, does this always increase the size of the statment’s truth set? 6. ( 3.2, Prob 6 ) Using s ( x, y, z ) to be the statement x = yz and t ( x, y ) to be the statement x < y , write a formal statement for the deFnition of the greatest common divisor of two numbers. 7. ( 3.2, Prob 10 ) Rewrite the following statement without any negations: It is not the case that there exists an integer n such that n > 0 and for all integers m > n , for every polynomial equation p ( x ) = 0 of degree m there are no real numbers for solutions. 1...
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