Monday, July 28, 2014

Truth Tables

published on: 2/28/2003

Contributing Teacher(s): Kathy Steinhoff

Subject Area: Math/Discrete Math- Graphs, Sets, Problem-Solving

Grade Range: High School (9-12)

Materials Needed:

  • Truth Tables and Answer Keys - Attached

Objective:

  1. This lesson is designed to teach students about formal logic using truth tables.

Instructional Strategy: Generating and Testing Hypothesis

Process Standards:

  • Goal 3.2 develop and apply strategies based on ways others have prevented or solved problems
  • Goal 3.5 reason inductively from a set of specific facts and deductively from general premises

Content Standards:

  • Mathematics 6. Discrete mathematics (such as graph theory, counting techniques, matrices)

    G.L.E.:

    Time Allowance: 2 or more days

    Description: Uses truth tables to decide the values of conditional statements.

    Comments: This would be appropriate for a Geometry course or possibly an honors course. The symbolism is probably too difficult for lower level students. Students that are interested in this sort of mathematics should be encouraged to pursue other lessons in the area of logic. There are several good resources available on the Internet. Some good keywords to search with are: logic, discrete mathematics, truth tables, venn diagrams, and Boolean logic.


    Classroom Component: Background

    A conditional statement is made up of a hypothesis, the if part, and a conclusion, the then part. For a conditional statement to be true, the conclusion must be true whenever the hypothesis is true. In formal logic, a conditional statement is also true if the hypothesis is false. In everyday situations we would probably consider such a statement irrelevant.

    Let's look at the following example:

    Conditional statement: If you clean your room then you can go to the movies.
    The hypothesis is: you clean your room
    The conclusion is: you can go to the movies

    Determining whether the conditional statement is true depends on if its components are true.

    • If you cleaned your room (T) and got to go to the movies (T), then the conditional is true (T).
    • If you cleaned your room (T) and did not get to go to the movies (F) then the conditional is false (F).
    • If you didn't clean your room (F) and got to go to the movies (T), then the conditional is true (T).
    • If you didn't clean your room (F) and did not get to go to the movies (F) then the conditional is true (T).

    These last two probably seem strange. Think of it this way, if your father made the original statement to you, under what conditions was he lying? The only time we know for sure he lied was if you cleaned your room and he wouldn't let you go. If you didn't clean your room, we don't know what he would have done so we have to assume he was telling the truth.

    In formal logic, the components, in the previous example cleaning your room and going to the movies, are irrelevant. The logic should be the same no matter what the components are. Therefore we commonly refer to the components as p and q. The logic symbolism goes much further in the previous example, we would have converted the conditional statement to if p then q and an even simpler conversion to p => q. We would then make a truth table to help us keep track of the possible truth values of this conditional.

    P Q P=>Q
    T T T
    T F F
    F T T
    F F T


    Lesson
    Day 1:

    After students are exposed to how to complete truth tables. Let them complete the "Basic Truth Tables" Worksheet. They probably need to review the following terms and symbolic representations.


    Term Symbol(s) Definition
    Term
    Symbol (s)
    Definition
    and
    ^
    Both have to be true for the compound statement to be true
    or
    v
    One or the other or both have to be true for the compound statement to be true
    Negation
    ~ or ¬
    The opposite truth value
    Conditional
    p=>q
    P implies q, stated as "if…then.."
    Converse
    q=>p
    The order of the hypothesis and conclusion is reversed
    Inverse
    ~p=>~q
    Both hypothesis and conclusion are negated
    Contrapositive
    ~q=>~p
    Hypothesis and conclusion are negated and reversed
    Biconditional
    pq
    When both a conditional and its converse are true; stated as "if and only if"


    After students have completed and understand the "Basic Truth Tables" worksheet, assign "Getting At The Truth" for homework. Suggest they use their "Basic Truth Tables" worksheet to help them.

    Day 2:

    Go over their homework to check for student understanding before moving on to more complicated problems. Also take time to review the "Basic Truth Tables" worksheet; have students label each truth table with the proper name and discuss truth tables that are equivalent. For example, conditionals and contrapositives have the same truth values. Lead them through a few of the problems from the "Challenging Truth Tables" worksheet and let them attempt the rest on their own or in groups. When they complete this, have them check their answers using the truth table generator at the following website.

    Truth table generator ~ http://sciris.shu.edu/~borowski/Truth/
    The truth table generator website takes you to Brian's Java Site and then if you scroll down to the bottom of the page there is a button for the truth table constructor.

    For those that finish early that day or for those that need more practice or will need to review later, suggest they look at this website.

    Truth table practice ~ http://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.html


    BASIC TRUTH TABLES

    Basic Truth Tables


    CHALLENGING TRUTH TABLES

    Challenging Truth Tables


    BASIC TRUTH TABLES

    ANSWER KEY

    Basic Truth Tables Answer Key


    CHALLENGING TRUTH TABLES

    ANSWER KEY

    Challenging Truth Tables Answer Key


    Getting At the Truth

    Use logic to draw conclusions concerning Jeanie's day of causing tr

    For additional information contact :
    Kathy Steinhoff
    Jefferson Jr. High
    Columbia 93
    (573) 214-3210
    EMAIL:
    ksteinho@columbia.k12.mo.us

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