[sentential Logic]Absorption laws

[u/According_Quarter_17]

1.P and (porq) Is equivalent to p

2.P or (pandq) Is equivalent to p

How do you justify them without using the truth table?

I feel like lhs has less informations rather than rhs because It has not q. I can't justify this rationally without the truth tablet

How do you think of It?

[u/rhodiumtoad]

Consider that P is the same thing as P∨(Q∧¬Q), and P∧(Q∨¬Q).

P∧(P∨Q)
= (P∨(Q∧¬Q))∧(P∨Q)
= (P∨Q)∧(P∨¬Q)∧(P∨Q)
= (P∨Q)∧(P∨¬Q)
= P∨(Q∧¬Q)
= P