EXAM CS250 2 CRAM GUIDE!
***THIS GUIDE IS A STUDENT CREATED DOCUMENT***
1.6: Definitions Program Verification: attempts to ensure that a computer program is correct. Correct: A program is correct if it behaves in accordance with its specifications. o However, this does not necessarily mean that the program solves the problem that it was intended to solve; the program's specifications may be at odds with or not address all aspects of a client's requirements. Program Validation: attempts to ensure that the program indeed meets the client's original requirements. Program Testing: seeks to show that particular input values produce acceptable output values. o "testing can prove the presence of errors but never their absence." o Except for rather simple programs, multiple tests that reveal no bugs do not guarantee that the code is bug-free, that there is not some error lurking in the code waiting to strike under the right circumstances. Proof of Correctness: uses the techniques of a formal logic system to prove that if the input variables satisfy certain specified predicates or properties, the output variables produced by executing the program satisfy other specified properties. Hoare Triple: {Q} P {R} (where P stands for program P, Q is the precondition and R is the postcondition) Conditional rule of inference: determines when a Hoare triple can be inserted in a proof sequence if siis a
conditional statement.
Assertions: predicates, often used in program verification methods that follow to the structure of a Hoare triple, that assert what is supposed to be true about the program variables at that point in the program. Examples: 1. (T/F) A provably correct program always gives the right answers to a given problem. 2. (T/F) If an assertion after an assignmnent is y > 4, then the precondition must be y 4. 3. (T/F) Proof of correctness involves careful development of test data sets. 4. (T/F) Using the conditional rule of inference in proof of correctness involves proving that two different Hoare triples are valid. 5. (T/F) the assertions used in proof of correctness can also be used as a program design aid before the program is written, and as program documentation. Answers: 1. False; It guarantees only that the output satisfies certain conditions, given that the input satisfies certain conditions. 2. False; Nothing much can be said about the precondition without knowing the assignment, but at any rate the strict inequality will not go away. 3. False; Program testing involves test data sets. 4. True 5. True Review problems 1.6.1, 1.6.8, 1.6.10 and 1.6.11 by using the homework solutions available at www.cis.uab.edu/ranelli 2.1: Definitions Inductive Reasoning: drawing a conclusion based on example. Deductive Reasoning: a method that aims to verify the truth or falsity of the conjecture. Counterexample: an example that the disproves conjecture. (Claim: Everyone hates 250. Counterexample: John Square absolutely loves 250. Axioms: Statements that are always true. o Example: Given two distinct points, there is exactly one line that contains them.
Proof by Exhaustion (Exhaustive proof): a proof where all possible cases have been exhausted. This is usually applicable where there are very few true cases (page 91 in the book). Direct Proof: a proof that assumes the hypothesis P and deduce the conclusion Q. Contrapositive (Proof by Contraposition): proving a proof by showing that the opposite of the end result implies the opposite of the beginning result. o This is valid because of the tautology: (P Q) (Q' P') Contradiction (Proof by Contradiction): proving the validity of a proof by finding a contradiction o Sometimes called an "indirect proof" o Uses the tautology: (P Q0) (P Q) (the symbol 0 means contradiction, or a case that is never true). Know theses informal proof methods and be able to decipher when each one is needed: o Inductive reasoning o Deductive reasoning o Proof by exhaustion: used with finite cases with few true outcomes o Direct proof o Proof by contraposition o Proof by contradiction o Serendipity: a fortuitous happening, or good luck!
Examples:
1. 2. 3. 4.
(T/F) A conjecture can never be proved merely by proving a finite number of cases. (T/F) A proof by contradiction of P Q begins by assuming both P and Q'. (T/F) In the staement of theorem "twice an odd integer is even," an existential quantifier is understood. (T/F) To prove the conjecture "If Laramie is the capital, then Wyoming is the state," it is sufficient to prove "if Wyoming is the state, then Laramie is the capital." 5. (T/F) To prove "A if and only if B" requires a proof of A B and a proof of B A. Answers:
1. False; A conjecture that only asserts something about a finite number of cases can be proved by proving all cases. 2. True 3. False; A universal quantifier is understood, because the formal statement of the theorem is
(x)(x is odd 2 * x is even). 4. False, the second statement is the converse of the first, not the contrapositive. 5. True
Review problems by using the homework solutions available at www.cis.uab.edu/ranelli and the problems in the back of the book. 2.2: Definitions Basis step (Basis): The first step in proving a claim through induction is establishing base a case, or P(1). o Establishing the truth for this statement then leads you to proving P(k + 1). o If you assume P(k) or P(k + 1) to be true, then they are inductive hypotheses. Just know the form/structure for proving a claim via inductive reasoning and be able to work problems. They are all fairly similar. Examples 1. (T/F) Induction is an appropriate proof technique for proving a statement about all the positive integers. 2. (T/F) The basis step of an inductive proof requires proving a property true for n = 1.
3. (T/F) If the truth of P(k + 1) depends on the truth of the other previous values besides P(k), then the second principle of induction should be used. 4. (T/F) The key to a proof by the first principle of induction is to see how the truth of P at the value k + 1 depends on the truth of P at value k. Answers 1. True 2. False; The basis step need not be n = 1. 3. True 4. True Be able to do these problems (it's the only thing in this section, so any test question is bound to solve or verify the validity of such problems)! They all follow a very structured format, the only thing that varies is the equation that you are trying to prove. Remember, the base case does not have to be 1! 2.3: I really don't know if we covered any of this (I am not very alert in class... ). But alas, be prepared. 2.4: Definitions (Know recursion stuff because Ranelli likes it!!!) Recursion: A sequence is defined recursively by explicitly naming the first value (or the first few values) in the
sequence and then defining later values in the sequence in terms of earlier values.
Recursive Definition: when a definition in which the item being defined appears as part of the definition Sequence: a sequence S is a list of objects that are enumerated in some order o Fibonacci Sequence: 0 1 1 2 3 5 8 13 21 34 55 89 etc. Know the multiplication, exponential, and factorial operations! Examples 1. (T/F) A sequence defined by(see below) contains the number 215. S(1) = 7 S(n) = 3S(n 1) + 2 for n 2 2. (T/F) Recursive algorithms are valuable primarily because they run more efficiently than iterative algorithms. Answers 1. True 2. False; they are valuable because they represent natural ways of thinking about certain problems, but they typically use more storage and perform more operations than a corresponding iterative program. 3.1: Definitions Set: A collection of objects. o Two sets are equal if they contain the same objects. o Sets can be finite or infinite. Important number abbreviations! o N: set of all nonnegative integers (includes 0) o Z: set of all integers o Q: set of all rational numbers o R: set of all real numbers o C: set of all complex numbers Braces {} are used to indicate a set.
Sets can be finite or infinite (this has to do with their countability) Empty Set (or ): a set with no values Subset ( ): A is a subset of B if for all x, (x A x B) Proper Subset ( ): A is a proper subset of B if there is at least one element of B that is not an element of A. Powerset: if for a set S, you can form a set whose elements are all of the subsets of S. o Basically, a power set is a set of sets. Cardinality: the number of elements in a set (denoted by |S|). Unary Operation (on set S): look at book examples: pages 192-195 Binary Operation (on set S): look at book examples: pages 192-195
Union: The union of set A and B, denoted by A B is the set that contains all elements in either set A or set B, i.e. A B = {x| xA or xB}. Intersection: The intersectionof set A and B, denoted by A B contain all elements that are common to both sets i.e. A B = {x| xA and xB} Cartesian Product: If A and B are subsets of S, then the cartesianproduct(cross product)of A and B denoted symbolically by A B is defined byA B = {(x,y) | xA and yB } The Cartesian product of 2 sets is the set of all combinations of ordered pairs that can be produced from the elements of both sets. Example, given 2 sets A and B, where A = {a, b, c} and B = {1, 2, 3}, theCartesian product of A and B can be represented asA B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}
Countable: This denotes a set that is finite, and therefore has a countable amount of elements. Uncountable: This denotes a set that is infinite, and has an uncountable amount of elements. Examples 1. (T/F) The empty set is a proper subset of every set. 2. (T/F) If A and B are disjoint sets, then (A B) u (B A) = A u B 3. (T/F) If a set has n elements, then its power set has 2n elements. 4. (T/F) If a binary operation on a set S is well-defined, then x y S for all x and y in S. Answers 1. False; It is not a proper subset of itself. 2. True 3. True 4. False; This is the closure property. Be sure to review book problems and homework problems for this section. I can assure you that you will have to tell whether or not sets are subsets of eachother, or something like Practice 6 on page 191! Read Sections 3.2, 3.3, and 3.4 just in case! We covered them briefly in class (but not on the homework), so be ready for anything. Also, review the binary homework and be able to convert numbers into various forms. I was going to put more actual example problems on here, but I ran out of time. Sorry!
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