Tautologies using metalogic

( Φ ∧  Ψ ) -> Φ and ( Φ ∧ Ψ ) -> φ are tautologies. Why? Since for the same reason that (p ∧ q) -> p as well as (p ∧ q) -> q are tautologies. If you do the truth tables:

(p ∧ q ) -> p

p q p ∧ q (p∧ q) -> p
T T T T
T F F T
F T F T
FF F T

ALL values are TRUE...thus it is a tautology.

Just do the same for (Φ ∧ Ψ ) -> Φ and ( Φ ∧  Ψ ) -> φ as they are also tautologies. Remember these symbols "Φ" and "Ψ" are just symbols that represent other symbols in metalogic. We can substitute (p ∧ q ) -> p and see that is a tautology as all values are T.

(So it is clear you get T for (p ∧ q) ->p by implication, since
p ->q means ¬p V q or here it, (p ∧ q) ->p, means ¬(p∧ q) V p )