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Study Guides > Mathematics for the Liberal Arts

Introduction

Let’s play a game!

  Tic-Tac-Toe game at a playground made of big yellow cylinders with black X's and O's printed on them.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 rules are so simple that kids as young as 3 or 4 can get the idea.   At first, a young child may play haphazardly, marking the grid without thinking about how the other player might respond.  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.   It’s not until about age 6 or so that children begin to strategize, looking at their opponent’s potential moves and responses.  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.     The logic involved can be fairly complex, especially for a young child.  For example, suppose it’s your turn (X’s), and the grid currently looks like this.  Where should you play? Tic-tac-toe game with two X's and two O's.   Your thought process (or what we call a logical argument) might go something like this:
  • It takes three in a row to win the game.
  • I cannot make three in a row no matter where I play on this turn.
  • If it were my opponent’s turn, then she could make three in a row by putting an O in the upper left corner.
  • If I don’t put my X in the upper left corner, then my opponent will have the opportunity to play there.
  • Therefore, I must put an X in the upper left corner.
Tic-Tac-Toe example, continued, with red X in the upper left corner.   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.  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.   Indeed, computers follow the rules of logic by design.  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.   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.  

Learning Objectives

Organize Sets and Use Sets to Describe Relationships
  • Describe memberships of sets, including the empty set, using proper notation, and decide whether given items are members and determine the cardinality of a given set
  • Perform the operations of union, intersection, complement, and difference on sets using proper notation
  • Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation
  • Be able to draw and interpret Venn diagrams of set relations and operations and use Venn diagrams to solve problems
  • Recognize when set theory is applicable to real-life situations, solve real-life problems, and communicate real-life problems and solutions to others
Introduction to Logic
  • Combine sets using Boolean logic, using proper notations
  • Use statements and conditionals to write and interpret expressions
  • Use a truth table to interpret complex statements or conditionals
  • Write truth tables given a logical implication, and it’s related statements – converse, inverse, and contrapositive
  • Determine whether two statements are logically equivalent
  • Use DeMorgan’s laws to define logical equivalences of a statement
Analyzing Arguments With Logic
  • Discern between an inductive argument and a deductive argument
  • Evaluate deductive arguments
  • Analyze arguments with Venn diagrams and truth tables
  • Use logical inference to infer whether a statement is true
  • Identify logical fallacies in common language including appeal to ignorance, appeal to authority, appeal to consequence, false dilemma, circular reasoning, post hoc, correlation implies causation, and straw man arguments
 

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