Question

I don't understand how ~(p v q) v ((p v q v r)^(~r v r)) can be equivalent to ~(p v q) v (p v q) v r and how you can just lose the brackets like that my teacher used something called the law of excluded middle I don't understand how ~(p v q) v ((p v q v r)^(~r v r)) can be equivalent to ~(p v q) v (p v q) v r and how you can just lose the brackets like that my teacher used something called the law of excluded middle @Mathematics

Answer 1

identity and associativity law. When I saw this I made ~r v r into T using inverse laws but I can't get simplify after that

Answer 2

\[(p \cup q \cup r)\cap(r\cup\sim r ) \iff (p \cup q \cup r) \]
Because
\[r \cup \sim r\]
is a tautology, i.e. it is by definition true. You can verify using a truth table that
\[(p \cup q) \cup r \iff (p \cup q \cup r)\]
if that was causing you difficulty. The law of excluded middle is the basis for deciding that
\[r \cup \sim r\]
is a tautology. The essential point of the thing is that if r is true, then ~r is not, or vice versa. Given any proposition r, either r is true or ~r is true, and so it follows that
\[r \cup \sim r\]
is true by default.

Answer 3

Oh I see (p v q v r )& T gives p v q v r

Answer 4

my only question is why is the brackets not after r anymore

Answer 5

It becomes (p v q) v r rather than (p v q v r)

Answer 6

Is there a law you can write for it? I can see that it's just a law

Answer 7

That would be the Associative Law, that says that
\[A \cup B \cup C \iff (A \cup B) \cup C \iff A \cup (B \cup C)\]

Answer 8

Oh okay thank you that's I was going to say. Thanks for your help!!

Answer 9

No problem :)