At first, a young child may play haphazardly, marking the grid without thinking about how the other player might respond. So [latex]\textrm{Q} =\overline{\textrm{A}} \cdot \overline{\textrm{B}}[/latex]. Logic and Set Theory. Therefore, I must put an X in the upper left corner. 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. All Rights Reserved. For all ages and levels. Primitive Concepts. Beautifully illustrated, low-priced Dover coloring on an amazing variety of subjects. Instead of truth values T and F, digital computers rely on two states, either on(1) or off(0). Bestselling series of coloring books for adults offers highly detailed illustrations on premium paper – relax and color. Indeed, computers follow the rules of logic by design. 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 mathematical logic compose the foundation of pure mathematics. 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. Search for: Putting It Together: Set Theory and Logic. There is also an OR gate. The rules are so simple that kids as young as 3 or 4 can get the idea. 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. So if the input (A) is 1, then the output (C) will be 0, and vice versa. First, both A and B are negated to obtain [latex]\overline{\textrm{A}}[/latex] and [latex]\overline{\textrm{B}}[/latex], respectively. For example, an AND gate has two input wires (A, B) and one output (C). Certain components called gates allow the computer to open or close circuits based on input., [latex]\overline{\textrm{A}}[/latex] (NOT A). 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. The logic involved can be fairly complex, especially for a young child. The rules are so simple that … Instead, let’s use Boolean logic and to find a simpler circuit equivalent to the one shown. 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 |, 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. Please go to In this module we’ve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. Corrected (1979) reprint of the W. H. Freeman & Co., San Francisco, 1963 edition. Moreover, numerical values can be represented by a string of 1’s and 0’s in what we call binary notation. Coloring books for adults and children. Mathematical Logic is a branch of mathematics which is mainly concerned with the relationship between “semantic” concepts (i.e. 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. Finally, the last expression corresponds to a circuit diagram with only two gates, an OR and a NOT. The gate along with its truth table shown below. Just think about the truth table for the operation [latex]\wedge[/latex], replacing T by 1 and F by 0. If it were my opponent’s turn, then she could make three in a row by putting an O in the upper left corner. 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). Your thought process (or what we call a logical argument) might go something like this: Because you are much more experienced than the typical 6 year-old child, I bet that you immediately saw where the X should be played, even without thinking through all of the details listed above. Finally, there is a gate whose output is the opposite state as its input. So, whether you are trying to find the winning Tic-Tac-Toe strategy, putting together a valid argument to convince fellow lawmakers to preserve important funding, or designing powerful computers to help solve complicated problems, logic is an essential part of our world.