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Math Help - First Absorption Law

  1. #1
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    First Absorption Law

    Verify the first Absorption Law by means of a truth table. I know that 0 = false and 1 = true, but I'm having trouble figuring out how to write a truth table for this.

    The first Absorption Law is:

    p v (p ^ q) = p

    --------------------------------------------------------------

    Here is what I gathered from examples in my book:

    I think the table will have 4 rows. On row 1, 'P' and 'Q' will both be true; on row 2, 'P' will be true and 'Q' will be false; on row 3, 'P' will be false and 'Q' will be true; and on row 4, 'P' and 'Q' will both be false.

    'P v (P & Q)' is true if either 'P' is true or 'P & Q' is true, so 'P v (P & Q)' is true whenever 'P' is true. Thus, 'P' and 'P v (P & Q)' are both true on lines 1 and 2.

    'P v (P & Q)' is false if 'P' and 'P & Q' are both false. Now, 'P & Q' is false if 'P' is false, so 'P v (P & Q)' is false whenever 'P' is false. Thus, 'P' and 'P v (P & Q)' are both false on lines 3 and 4.

    Thus, on each line of the truth table, the truth value that 'P' has on that line is the same as the truth value that 'P v (P & Q)' has on that line. Hence, 'P' and 'P v (P & Q)' are logically equivalent: they are true under exactly the same conditions.

    How would you write this in a truth table?
    Last edited by rhymin; March 24th 2013 at 01:53 PM.
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  2. #2
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    Re: First Absorption Law

    Write a table with four rows as you described and at least 3 columns. Two columns are for values of p and q, and the third one is for the truth value of p v (p ^ q) for the given p and q. If you'd like, you may have an another column for the intermediate value of p ^ q.

    Whether you have the fourth (or fifth) column depends on whether = in p v (p ^ q) = p is a connective (equivalence, or biconditional), just like v or ^, or whether it is a meta-level equality saying that the truth values of two formulas are always equal. If it is a connective, then the last columns must list the truth values of the whole formula p v (p ^ q) = p. (In this case, columns for subformulas p ^ q and p v (p ^ q) are optional.) This column is expected to have only T. If = is a meta-level equality, then you just inspect the columns for p and for p v (p ^ q) and observe that they are identical.
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  3. #3
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    Re: First Absorption Law

    So I think I have the first 2 columns and last column correct, but I'm not sure about the 3rd.
    p
    1
    1
    0
    0

    q
    1
    0
    1
    0

    p^q
    1
    0?
    0?
    0?

    p v (p ^ q)
    1
    1
    0
    0

    Is this correct, or is my 3rd column "p ^ q" wrong?
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  4. #4
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    Re: First Absorption Law

    Quote Originally Posted by rhymin View Post
    So I think I have the first 2 columns and last column correct, but I'm not sure about the 3rd.

    Code:
    p  q  p ^ q  (p v (p ^ q))
    1  1    1         1
    1  0    0?        1
    0  1    0?        0
    0  0    0?        0
    Is this correct, or is my 3rd column "p ^ q" wrong?
    This truth table is correct. Concerning your question about the truth values of the conjunction p ^ q, don't you have a textbook or lecture notes that have the truth table for conjunction?

    Hint: You can place your table between the [code]...[/code] tags to preserve alignment, as in the quote above. (Click "Reply With Quote" button to see the code.) Or check the code for this post to see the forum tags for creating tables.
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  5. #5
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    Re: First Absorption Law

    Thank you for that information!
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