Question

9. Use truth tables to prove that the following are equivalent:

Answer 1

9.
(a)\[\begin{array}{|c|c|c|c|c|c|}\hline\phi&\psi&\neg\psi&\phi\Rightarrow\psi&\neg(\phi\Rightarrow\psi)&\phi\wedge(\neg\psi)\\\hline T&T&F&T&F&F\\ T&F&T&F&T&T\\F&T&F&T&F&F\\F&F&T&T&F&F\\\hline \end{array}\]

(b) \[\begin{array}{|c|c|c|c|c|c|c|c|}\hline\phi&\psi&\theta&\psi\wedge\theta&\phi\Rightarrow(\psi\wedge\theta)&\phi\Rightarrow\psi&\phi\Rightarrow\theta&(\phi\Rightarrow\psi)\wedge(\phi\Rightarrow\theta)\\\hline T&T&T&T&T&T&T&T\\T&T&F&F&F&T&F&F\\T&F&T&F&F&F&T&F\\T&F&F&F&F&F&F&F\\F&T&T&T&T&T&T&T\\F&T&F&F&T&T&T&T\\F&F&T&F&T&T&T&T\\F&F&F&F&T&T&T&T\\\hline\end{array}\]

(c) \[\begin{array}{|c|c|c|c|c|c|c|c|}\hline\phi&\psi&\theta&\phi\vee\psi&(\phi\vee\psi)\Rightarrow\theta&\phi\Rightarrow\theta&\psi\Rightarrow\theta&(\phi\Rightarrow\theta)\wedge(\psi\Rightarrow\theta)\\\hline T&T&T&T&T&T&T&T\\T&T&F&T&F&F&F&F\\T&F&T&T&T&T&T&T\\T&F&F&T&F&F&T&F\\F&T&T&T&T&T&T&T\\F&T&F&T&F&T&F&F\\F&F&T&F&T&T&T&T\\F&F&F&F&T&T&T&T\\\hline\end{array}\]

Answer 2

is 9 significant?

Answer 3

well i think have done question nine ,

10. Verify the equivalences in (b) and (c) in the previous question by means of a logical argument. (So, in the case of (b), for example, you must show that assuming φ and deducing ψ ∧ θ is the same as both deducing ψ from φ and θ from φ.)

Answer 4

it is 10 im having trouble with

Answer 5

The equation editor is not really suited for logical reasoning alas

Answer 6

( A V B ) -> C
¬(A V B ) V C //definition of the ->
¬A ^ ¬B V C //deMorgan

( A -> C ) ^( B -> 0 )
¬A V C ^ ¬B V 0 //definition of the ->
¬A V C ^ ¬B // absorb the 0
¬A ^ ¬B V C

QED

Answer 7

(A V B ) -> C
¬( A V B ) V C //definition
( ¬A ^ ¬ B ) V C //deMorgan
¬A V C ^ ¬B V C //distributive

( A -> C ) ^ ( B -> C )
¬A V C ^ ¬B V C //definition

Answer 8

hmm

Answer 9

hmm? It's correct I assure you

Answer 10

10. Verify the equivalences in (b) and (c) in the previous question by means of a logical argument.


you need to verify b) and c) with an example is it ?