hey friends,

please help me out with this question! thanks lots! (Happy)

show that (p→q)∧(p→r) and p→(q∧r) is logically equivalent

thanks in advance!

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- Aug 30th 2011, 07:48 AMllapkanpropositional logic
hey friends,

please help me out with this question! thanks lots! (Happy)

show that (p→q)∧(p→r) and p→(q∧r) is logically equivalent

thanks in advance!

- Aug 30th 2011, 08:06 AMPlatoRe: propositional logic
- Aug 30th 2011, 09:52 AMHallsofIvyRe: propositional logic
Another way to do this is to construct a "truth table". p, q, and r can each be "T" or "F" so there are $\displaystyle 2^3= 8$ cases.

If p= q= r= T, then $\displaystyle p\to q$ is true and $\displaystyle p\to r$ is true so $\displaystyle (p\to q)\wedge(\p\to r)$ is true. Also then [tex]q\wedge r[/itex] is true so $\displaystyle p\to (q\wedge r)$ is true. Since they are both true $\displaystyle (p\to q) \wedge (p\to r)= p\to (q \wedge r)$ in this case.

If p= F, which includes 4 of the 8 cases, each of $\displaystyle p\to q$, $\displaystyle p\to r$, and $\displaystyle p\to(q\wedge r)$ is true because "$\displaystyle F\to x$" is true no matter what x is.

Now show they are the same in the other 3 cases.