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set theory and logic

If it were my opponent’s turn, then she could make three in a row by putting an O in the upper left corner. © 2013 Dover Publications. Coloring books for adults and children. In this module we’ve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Almost everyone knows the game of Tic-Tac-Toe, in which players mark X’s and O’s on a three-by-three grid until one player makes three in a row, or the grid gets filled up with no winner (a draw). |, The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise, Undecidable Theories: Studies in Logic and the Foundation of Mathematics, Sets, Sequences and Mappings: The Basic Concepts of Analysis. All Rights Reserved. Finally, the last expression corresponds to a circuit diagram with only two gates, an OR and a NOT. Little did he realize that his system of “and,” “or,” and “not” operations would one day transform the world by ushering in the Digital Revolution and modern day computing. Check out the newest Creative Haven® coloring releases! The rules are so simple that kids as young as 3 or 4 can get the idea. So [latex]\textrm{Q} =\overline{\textrm{A}} \cdot \overline{\textrm{B}}[/latex]. Logic and Set Theory. What is the connection between logic and computers? This is because a computer consists of many circuits, which are electrical pathways that can either be closed to allow the current to flow, or open to break the connection. Module 6: Set Theory and Logic. That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t define, but which we assume satisfy some Multiplication seems to be a natural interpretation of AND when applied to the values 0 and 1. Set theory and mathematical logic compose the foundation of pure mathematics. In fact when George Boole (1815-1864) first developed symbolic logic (or Boolean logic), he had the idea that his system could be used by lawyers, philosophers, and mathematicians alike to help put convoluted arguments on a firmer footing. Just think about the truth table for the operation [latex]\wedge[/latex], replacing T by 1 and F by 0. Primitive Concepts. Using the axioms of set theory, we can construct our universe of discourse, beginning with the natural numbers, moving on with sets and functions over the natural numbers, integers, rationals and real numbers, and eventually developing the transfinite ordinal and cardinal numbers. Those expressions in turn feed into the AND gate. Like logic, the subject of sets is rich and interesting for its own sake. Logic and Set Theory. You have encountered “not” as the logical expression [latex]\sim\!\textrm{A}[/latex], but  the usual notation in computer science for NOT A is [latex]\overline{\textrm{A}}[/latex]. For example, an AND gate has two input wires (A, B) and one output (C). Search for: Putting It Together: Set Theory and Logic. Instead of truth values T and F, digital computers rely on two states, either on(1) or off(0). Let’s translate the diagram into a Boolean expression. Instead, let’s use Boolean logic and to find a simpler circuit equivalent to the one shown. The logic involved can be fairly complex, especially for a young child. Bestselling series of coloring books for adults offers highly detailed illustrations on premium paper – relax and color. You may recognize the expression as one side of De Morgan’s Law. Certain components called gates shunt electricity in various ways throughout the circuitry of the computer, allowing it to perform whatever procedures it is programmed to do. mathematical objects) and “syntactic” concepts (such as formal languages, formal deductions and proofs, and computability). Certain components called gates allow the computer to open or close circuits based on input. Why understand set theory and logic applications? A “1” would signify a closed circuit while a “0” represents an open circuit. https://commons.wikimedia.org/wiki/File:George_Boole_color.jpg, [latex]\overline{\textrm{A}}[/latex] (NOT A). The gate along with its truth table shown below. At first, a young child may play haphazardly, marking the grid without thinking about how the other player might respond. Where should you play? Therefore, I must put an X in the upper left corner. For example, suppose it’s your turn (X’s), and the grid currently looks like this. In fact, if you have played a fair number of Tic-Tac-Toe games in your childhood, then there are neural pathways in your brain that are hard-wired for Tic-Tac-Toe logic, just like a computer might be hard-wired to complete certain routine tasks. Set Theory and Logic is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics.Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. It’s not until about age 6 or so that children begin to strategize, looking at their opponent’s potential moves and responses. Corrected (1979) reprint of the W. H. Freeman & Co., San Francisco, 1963 edition. We will return to sets as an object of study in chapters 4 and 5 . There is also an OR gate. The rules are so simple that … In this module we’ve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions.The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. First, both A and B are negated to obtain [latex]\overline{\textrm{A}}[/latex] and [latex]\overline{\textrm{B}}[/latex], respectively. The given circuit has three gates. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. In mathematics, the notion of a set is a primitive notion. https://pixabay.com/en/tic-tac-toe-game-tick-tack-toe-355090. This operation, which corresponds to the logical expression [latex]A \vee B[/latex], is often interpreted as a kind of addition (A OR B = A + B), however it’s not a perfect analogy because [latex]1+1=1[/latex] in Boolean logic. Almost everyone knows the game of Tic-Tac-Toe, in which players mark X’s and O’s on a three-by-three grid until one player makes three in a row, or the grid gets filled up with no winner (a draw). This is called the NOT gate. For all ages and levels. Again, two inputs A and B determine the output C, however this time C = 1 if and only if either A or B (or both) is equal to 1. Can you find a circuit with only two gates that produces exactly the same output (Q) for all choices of input (A, B)? Traditionally, the AND operation is written like multiplication; that is, A AND B = AB. Mathematical Logic is a branch of mathematics which is mainly concerned with the relationship between “semantic” concepts (i.e. Therefore, there is an equivalence, [latex](\sim\!\textrm{A})\wedge(\sim\!\textrm{B})= \;\sim\! (\textrm{A} \vee \textrm{B}) = \overline{\textrm{A} + \textrm{B}}[/latex]. For example, the child might eagerly make two in a row but fail to see that his older sister will be able to complete three in a row on her next turn. The child begins to use systematic reasoning, or what we call logic, to decide what will happen in the game if one move is chosen over another. Indeed, computers follow the rules of logic by design.

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