Finite Automata

Mathematical Logic Proof Question with quantifiers ~ discrete math
Since I can’t paste MathType format in here, this is how to decode the question:

“@” is the universal quantifier.

“#” is the existential quantifier.

“~” is negation

“^” is a conjunction.

“v” is a disjunction.

“->” is a conditional.

Please PROVE the following:

1) (@x)((F(x) v G(x)) -> ~H(x))

2) (#x)(F(x) ^ ~Z(x))

3) Given: (@x)(F(x) -> T(x))

Prove: (#x)(T(x) ^ ~H(x))
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