Move the negation inside as far as it will go: ~(∃x∈S)(∀z∈T)( ∃y∈U) x ≥ sin z ⇒ y < x^2 + z^2
You know how to bring a negation inside a quantifier?
Keep three things in mind here: To bring a negation inside an existence quantifier, change it to a universal quantifier and then negate the inside. To bring a negation inside a universal quantifier, change it to an existence quantifier and then negate the inside. To negate a statement of the form p => q, it would become p ^ ~q (p and not q)
So here goes... \[\sim(\exists x \in S)(\forall z \in T)(\exists y \in U)\ x \ge \sin z \rightarrow y < x^{2} + z^{2}\]First, bring the negation inside the first existence quantifier... \[(\forall x \in S)\sim(\forall z \in T)(\exists y \in U)\ x \ge \sin z \rightarrow y < x^{2} + z^{2}\]Next, bring the negation inside the universal quantifier... \[(\forall x \in S)(\exists z \in T)\sim(\exists y \in U)\ x \ge \sin z \rightarrow y < x^{2} + z^{2}\]Then, bring the negation inside the second existence quantifier... \[(\forall x \in S)(\exists z \in T)(\forall y \in U)\sim (x \ge \sin z \rightarrow y < x^{2} + z^{2})\] Note how the quantifier changed everytime the negation crossed it. Now it is down to negating something of the form p => q, which we know now to be p ^ ~q (Since p -> q is equivalent to ~p v q, and the negation of ~p v q is p ^ ~q) Therefore, it is down to... \[(\forall x \in S)(\exists z \in T)(\forall y \in U)x \ge \sin z\ AND \ y \ge x^{2} + z^{2}\] (Remember that the negation of less than is greater than OR equal to...) :D
Thanks guys, I just didn't know whether I should keep going after bringing the negation inside the quantifiers :)
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