# 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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