Concrete abstractions : an introduction to computer science by Max Hailperin

By Max Hailperin

This article covers the fundamentals of programming and knowledge buildings, and offers first-time machine technological know-how scholars the chance not to merely write courses, yet to turn out theorems and learn algorithms to boot.

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Similarly, it would be desirable to be able to join two images side by side as well as stacking them on top of one another. a. Define procedures half-turn and quarter-turn-left that do as their names suggest. Both procedures take a single argument, namely, the image to turn. You will naturally need to use the built-in procedure quarter-turn-right. b. Define a procedure side-by-side that takes two images as arguments and creates a composite image having the first image on the left and the second image on the right.

12 Any positive integer i can be expressed as i ϭ 2n k, where k is odd, that is, as a power of 2 times an odd number. We call n the exponent of 2 in i. For example, the exponent of 2 in 40 is 3 (because 40 ϭ 23 5) whereas the exponent of 2 in 42 is 1. If i itself is odd, then n is zero. If, on the other hand, i is even, that means it can be divided by 2. Write a procedure for finding the exponent of 2 in its argument. 4 An Application: Custom-Sized Quilts At the end of the previous chapter we made some quilts by pinwheeling basic blocks.

2 Write a similarly detailed proof of the factorial procedure’s correctness. What are the permissible argument values for which you should show that it works? Proving is also useful when you are trying to debug a procedure that doesn’t work correctly, that is, when you are trying to figure out what is wrong and how to fix it. For example, look at the incorrect version of square given earlier. If we were trying to prove that this works by induction, the base case and the inductive hypothesis would be exactly the same as in the proof above.

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