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    Discrete Math: Truth Tables

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    1) Construct the Truth Table for each of the following Boolean expressions:

    Are they equivalent expressions? Are they tautologies? Contradictions?

    2) Find a Boolean expression involving x y which produces the following table:

    3) Consider a statement of the form "if A then (B and C)". Assume you wish to disprove it. Then you need to provide a counterexample, that is, you need to exhibit an instance for which
    a) B and C are false
    b) A is true and both B and C are false
    c) B is false or C is false
    d) A is true and B is false, or A is true and C is false
    e) A is false, B is false, and C is false

    Indicate which of the above is the correct answer.

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    Solution Preview

    Please see the attached files.

    We use the equivalency:
    And two De-Morgan rules:
    The truth tables for the basic operations:

    F F F F
    F T T F
    T F T F
    T T T T

    The first expression can be converted using De-Morgan rule (b):


    The truth table is:

    F F T F
    F T F F
    T F T T
    T T F F

    The expression is simplified:
    This is a simple OR operation, so the truth table is:

    F F F
    F T T
    T F T
    T T T

    The expression can be simplifies using De-Morgan rule (a):
    This is a simple NAND operator. It ...

    Solution Summary

    The expert examines truth tables in discrete mathematics. Counterexamples are provided.