Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only negations, conjunctions and disjunctions: a) ∃𝑥𝑃(𝑥) b) ∀𝑥𝑃(𝑥) c) ∀𝑥((𝑥 ≠ 1) → 𝑃(𝑥)) d) ∃𝑥((𝑥 ≥ 0) ∧ 𝑃(𝑥)) e) ∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥((𝑥 < 0) → 𝑃(𝑥))

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a) $∃xP(x)=P(-5)\lor P(-3)\lor P(-1)\lor P(1)\lor P(3)\lor P(5)$

b) $∀xP(x)=P(-5)\land P(-3)\land P(-1)\land P(1)\land P(3)\land P(5)$

$∀𝑥((𝑥 ≠ 1) → 𝑃(𝑥))=∀𝑥(\neg(x\neq 1)\lor P(x))=(x=1)\lor P(1)$

$∃𝑥((𝑥 ≥ 0) ∧ 𝑃(𝑥))=((x=1)\land P(1))\lor ((x=)\land P(3))\lor ((x=5)\land P(5))$

$∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥((𝑥 < 0) → 𝑃(𝑥)) =∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥(\neg(𝑥 < 0) \lor 𝑃(𝑥))=$

$=∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥((𝑥 > 0) \lor 𝑃(𝑥))=$

$=∃𝑥(￢𝑃(𝑥)) ∧ (((𝑥 =1) \lor 𝑃(1))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))=$

$=(￢𝑃(-5)) ∧ (((𝑥 =1) \lor 𝑃(1))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))\lor$

$\lor (￢𝑃(-3)) ∧ (((𝑥 =1) \lor 𝑃(𝑥))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))\lor$

$\lor (￢𝑃(-1)) ∧ (((𝑥 =1) \lor 𝑃(𝑥))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))\lor$

$\lor (￢𝑃(1)) ∧ (((𝑥 =1) \lor 𝑃(𝑥))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))\lor$

$\lor (￢𝑃(3)) ∧ (((𝑥 =1) \lor 𝑃(𝑥))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))\lor$

$\lor (￢𝑃(5)) ∧ (((𝑥 =1) \lor 𝑃(𝑥))\land ((𝑥 =3) \lor 𝑃(3))\land ((𝑥 =5) \lor 𝑃(5)))$