Answer to Question #90135 in Discrete Mathematics for Rahim Ansari

Question #90135
Show that the following argument form is valid.

p --> q
q --> r

∴ p --> r
1
Expert's answer
2019-05-24T09:15:07-0400

We have to prove (p-->q) and (q-->r) implies (p-->r)

"((p\\to q)\\land(q\\to r))\\to(p\\to r)\\equiv"


"using(a\\to b)\\equiv(\\neg a\\lor b)"

"\\equiv\\neg((p\\to q)\\land(q\\to r))\\lor(p\\to r)\\equiv"

"using \\ \\neg(a\\land b)\\equiv(\\neg a\\lor \\neg b)"

"(\\neg(p\\to q)\\lor\\neg(q\\to r))\\lor(p\\to r)\\equiv"

"using(a\\to b)\\equiv(\\neg a\\lor b)"

"\\neg(\\neg p\\lor q)\\lor\\neg(\\neg q\\lor r)\\lor(\\neg p\\lor r)\\equiv"

"using \\neg(a\\lor b)\\equiv(\\neg a\\land \\neg b)"

"(\\neg\\neg p\\land \\neg q)\\lor(\\neg\\neg q\\land\\neg r)\\lor(\\neg p\\lor r)\\equiv"

"using \\neg\\neg a\\equiv a"

"( p\\land \\neg q)\\lor( q\\land\\neg r)\\lor\\neg p\\lor r\\equiv"

"using \\ commutative \\ law"

"( p\\land \\neg q)\\lor\\neg p\\lor( q\\land\\neg r)\\lor r\\equiv"

"using\\ associative\\ law\\"

"(( p\\land \\neg q)\\lor\\neg p)\\lor(( q\\land\\neg r)\\lor r)\\equiv"

"using\\ distributive\\ law"

"(( p\\lor\\neg p)\\land (\\neg q\\lor\\neg p))\\lor(( q\\lor r)\\land(\\neg r\\lor r))\\equiv"

"using\\ a\\lor \\neg a \\equiv T"

"(T\\land (\\neg q\\lor\\neg p))\\lor(( q\\lor r)\\land T)\\equiv"

"using \\ T\\land a\\equiv a"

"(\\neg q\\lor \\neg p)\\lor(q\\lor r)\\equiv"

"using\\ associative\\ law"

"(\\neg q\\lor q)\\lor(\\neg p\\lor r)\\equiv"

"using\\ a\\lor \\neg a \\equiv T"

"T\\lor(\\neg p\\lor r)\\equiv"

"using\\ T\\lor a\\equiv T"

"T"

Q.E.D.


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