Math

Construct a truth table for the statement.

(p ↔ q) → p

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asked by Jen
  1. Make a table for such that each column represents a value:
    (p <-> q) -> p
    T T T T T
    T
    !@#$%^&T
    1 3 2 4 1
    The last row represents the order in which the values are assigned.
    When you complete the table, you could post it for verification.

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  2. T T T T T
    T F F T T
    F T T F F
    T T T F !@#$%^&
    ??

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    posted by Jen
  3. There are some corrections to make, starting from p (first column) and q (third column). Sorry that the first column was wrong, it should have read:
    (p <-> q) -> p
    T T T T T
    T ? F
    !@#$%^&T
    !@#$%^&F
    1 3 2 4 1
    We usually give
    p [TTFF] and
    q [TFTF] in order to have all possible combinations.
    The "<->" operator is true when both p and q are true, or both are false,
    and "m->n" is false if and only if m is true and n is false.

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  4. i am still so confused.... :(

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    posted by Jen
  5. Work with a simple table:

    The biconditional p<->q is true if and only if both p and q are true, or both p and q are false.

    p q p<->q
    T T T
    T F F
    F T F
    F F T

    Now from this truth table, you will make another one for the conditional which is false if and only if p is true and q is false.
    p q p->q
    T T T
    T F F
    F T T
    F F T

    Now combine the two results to find
    (p <-> q) -> p,
    i.e. replace (p <-> q) with the results from the first table, and p remains [TTFF].
    Figure out the result of the conditional according to the second table.

    (p<->q) p (p<->q)->p
    T T ?
    F T ?
    F !@#$%^&
    T F ?

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  6. Jen,
    If you are still confused at this point, it may do you good to back up and study the truth tables for the basic logic operators:
    ∧, ∨ → ↔ ¬.

    Once you have mastered them, the truth tables of compound propositions would be much easier to do.

    Remember that you will soon be working with simplification of compound propositions by a dozen of known identities, which require the thorough knowledge of the simple logic operators.

    For the current problem, you will need the following:

    p q p<->q
    T T T
    T F F
    F T F
    F F T

    p q p->q
    T T T
    T F F
    F T T
    F F T

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  7. p q p <-> q (p<->q)->p
    T T T T
    T F F T
    F T F T
    F F T !@#$%^&?

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    posted by Jen
  8. Excellent, all correct!

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