329-16-17-PS1-SOLNS.pdf - UNIVERSITY OF TORONTO DEPARTMENT...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT138H1 - INTRODUCTION TO PROOFS – FALL 2016 PROBLEM SET #1. ANSWERS WITH BRIEF EXPLANATIONS. 1. This problem relates to an example solved in class during our first lecture. How many rectangles are there in an “expanded” 8 × 10 checkerboard? Answer: There are exactly 1980 rectangles in an “expanded” 8 × 10 checkerboard. In effect, without loss of generality, we can assume that the checkerboard consists of 8 rows and 10 columns. Suppose now that for each of our rectangles, we use 𝑎𝑎 and 𝑏𝑏 to denote the dimensions of the rectangle, with 𝑎𝑎 as its number of rows and 𝑏𝑏 as its number of columns. Clearly, for any 𝑎𝑎 and 𝑏𝑏 such that 𝑎𝑎 ∈ {1,2, … ,8} and 𝑏𝑏 ∈ {1,2, … ,10} there are some rectangles whose dimensions are 𝑎𝑎 and 𝑏𝑏 . Proceeding as we did in class when solving the related problem, we can conclude that for each of the 80 possible values of ( 𝑎𝑎 , 𝑏𝑏 ) , the number 𝑅𝑅 𝑎𝑎 , 𝑏𝑏 of rectangles with dimensions 𝑎𝑎 × 𝑏𝑏 in our 8 × 10 checkerboard is 𝑅𝑅 𝑎𝑎 , 𝑏𝑏 = (9 − 𝑎𝑎 )(11 − 𝑏𝑏 ) . Therefore, if we use 𝑅𝑅 to denote the total number of rectangles in the “expanded” 8 × 10 checkerboard, we finally have 𝑅𝑅 = (9 − 𝑎𝑎 )(11 − 𝑏𝑏 ) 10 𝑏𝑏=1 8 𝑎𝑎=1 = ( (9 − 𝑎𝑎 ) 8 𝑎𝑎=1 )( (11 − 𝑏𝑏 ) 10 𝑏𝑏=1 ) = 8 × 9 2 � � 10 × 11 2 = 1980 . 2. Suppose that the given truth table is used to define the logical connective . a) Is ( 𝑹𝑹 𝑸𝑸 ) ( 𝑹𝑹 𝑸𝑸 ) 𝑷𝑷� a tautology, a contradiction or neither? b) Is ( 𝑸𝑸 𝑸𝑸 ) ( 𝑷𝑷 𝑹𝑹 ) ( 𝑹𝑹 𝑷𝑷 ) a valid consequence of the propositions 𝑷𝑷 ( 𝑸𝑸 𝑹𝑹 ) and 𝑹𝑹 ( 𝑷𝑷 𝑸𝑸 ) ? Answers: a) The proposition ( 𝑹𝑹 𝑸𝑸 ) ( 𝑹𝑹 𝑸𝑸 ) 𝑷𝑷� is neither a tautology nor a contradiction. From the definition of , we can see that ( 𝑹𝑹 𝑸𝑸 ) ( 𝑹𝑹 𝑸𝑸 ) 𝑷𝑷� is true whenever 𝑸𝑸 and 𝑹𝑹 are both true. However, ( 𝑹𝑹 𝑸𝑸 ) ( 𝑹𝑹 𝑸𝑸 ) 𝑷𝑷� is false if the truth values of 𝑷𝑷 , 𝑸𝑸 and 𝑹𝑹 are F. F and T, respectively.
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