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- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/04%3A_Looking_Beyond/4.01%3A_Further_ReadingThe books by Christian & Griffiths, Hofstadter and Smullyan are on recreational maths. Hofstadter’s is a winner of the Pulitzer Prize. The other titles are textbooks; Hammack’s book is an Open Textboo...The books by Christian & Griffiths, Hofstadter and Smullyan are on recreational maths. Hofstadter’s is a winner of the Pulitzer Prize. The other titles are textbooks; Hammack’s book is an Open Textbook, just like this book is. Algorithms to Live By: The Computer Science of Human Decisions. New York, NY: Henry Holt and Co. Geneva, NY: Hobart and William Smith Colleges. Logic and Discrete Mathematics. Book of Proof, 3rd edition. and Ryan, M. (2004). What is the Name of This Book?
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/01%3A_Logic
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/01%3A_Logic/1.02%3A_Boolean_AlgebraHe worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854). “No general method for the solution of questions in the theory of ...He worked in the fields of differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854). “No general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.”
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/02%3A_Proof/2.05%3A_Mathematical_InductionIt’s like a line of dominos, lined up and ready to fall, one after the next.3 Since P(0) and P(0) → P(1) are true, we can apply the rule of modus ponens to conclude that P(1) is true. Stated formally,...It’s like a line of dominos, lined up and ready to fall, one after the next.3 Since P(0) and P(0) → P(1) are true, we can apply the rule of modus ponens to conclude that P(1) is true. Stated formally, the principle of mathematical induction says that if we can prove the statement P(0) ∧ (∀k (P(k) → P(k + 1)), then we can deduce that∀n P(n) (again, with N as the domain of discourse).
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/01%3A_Logic/1.01%3A_Propositional_Logic/1.1.06%3A_Implications_in_EnglishIn English, p → q is often expressed as ‘if p then q’. For example, if p represents the proposition “Karel Luyben is Rector Magnificus of TU Delft” and q represents “Prometheus is blessed by the gods”...In English, p → q is often expressed as ‘if p then q’. For example, if p represents the proposition “Karel Luyben is Rector Magnificus of TU Delft” and q represents “Prometheus is blessed by the gods”, then p → q could be expressed in English as “If Karel Luyben is Rector Magnificus of TU Delft, then Prometheus is blessed by the gods.” In this example, p is false and q is also false.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/02%3A_Proof/2.04%3A_Proof_by_ContradictionA sequence of statements that can be proved from those assumptions, and suppose that we derive a statement that we know to be false. When the laws of logic are applied to true statements, the statemen...A sequence of statements that can be proved from those assumptions, and suppose that we derive a statement that we know to be false. When the laws of logic are applied to true statements, the statements that are derived will also be true. If we derive a false statement by applying rules of logic to a set of assumptions, then at least one of the assumptions must be false. This observation leads to a powerful proof technique, which is known as proof by contradiction.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/01%3A_Logic/1.01%3A_Propositional_Logic/1.1.01%3A_PropositionsRemember that when I say something like “Let p be a proposition”, I mean “For the rest of this discussion, let the symbol p stand for some particular statement, which is either true or false (although...Remember that when I say something like “Let p be a proposition”, I mean “For the rest of this discussion, let the symbol p stand for some particular statement, which is either true or false (although I am not at the moment making any assumption about which it is).” The discussion has mathematical generality in that p can represent any statement, and the discussion will be valid no matter which statement it represents.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/02%3A_Proof/2.07%3A_Application_-_Recursion_and_InductionIn a computer program, a subroutine is a named sequence of instructions for performing a certain task. A typical way to organize a program is to break down a large task into smaller, simpler subtasks ...In a computer program, a subroutine is a named sequence of instructions for performing a certain task. A typical way to organize a program is to break down a large task into smaller, simpler subtasks by calling subroutines to perform each of the subtasks. Recursion is appropriate when a large task can be broken into subtasks where some or all of the subtasks are smaller, simpler versions of the main task.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/03%3A_Sets_Functions_and_Relations/3.03%3A_Programming_with_SetsOn a computer, all pieces of data are represented, ultimately, as strings of zeros and ones. How can sets be represented as strings of zeros and ones? And once we have represented sets on a computer, ...On a computer, all pieces of data are represented, ultimately, as strings of zeros and ones. How can sets be represented as strings of zeros and ones? And once we have represented sets on a computer, how do we program with them? In this section we go into detail about representing and computing with sets. You will find that there is again quite some overlap with your study materials from Computer Organ- isation however, as we discuss different numeric systems.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/01%3A_Logic/1.06%3A_LogicIn the example, the predicate is the phrase “is a city”. Once we start working with predicates, we can create propositions using quantifiers like ‘all’, ‘some’, and ‘no’. For example, working with the...In the example, the predicate is the phrase “is a city”. Once we start working with predicates, we can create propositions using quantifiers like ‘all’, ‘some’, and ‘no’. For example, working with the predicate “has a university” we can move from simple propositions like “Delft has a university” to “All cities have a university” or to “No city has a university” or to the rather more realistic “Some cities have a university”.
- https://eng.libretexts.org/Bookshelves/Computer_Science/Programming_and_Computation_Fundamentals/Delftse_Foundations_of_Computation/02%3A_ProofOf course, mathematics achieves this certainty by restricting itself to an artificial, mathematical world, and its application to the real world does not carry the same degree of certainty.Within the ...Of course, mathematics achieves this certainty by restricting itself to an artificial, mathematical world, and its application to the real world does not carry the same degree of certainty.Within the world of mathematics, consequences follow from assumptions with the force of logic, and a proof is just a way of pointing out logical consequences.
