Solution to Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only … - Sikademy
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Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only negations, conjunctions and disjunctions: a) βˆƒπ‘₯𝑃(π‘₯) b) βˆ€π‘₯𝑃(π‘₯) c) βˆ€π‘₯((π‘₯ β‰  1) β†’ 𝑃(π‘₯)) d) βˆƒπ‘₯((π‘₯ β‰₯ 0) ∧ 𝑃(π‘₯)) e) βˆƒπ‘₯(ᅭ𝑃(π‘₯)) ∧ βˆ€π‘₯((π‘₯ < 0) β†’ 𝑃(π‘₯))

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a)Β βˆƒxP(x)=P(-5)\lor P(-3)\lor P(-1)\lor P(1)\lor P(3)\lor P(5)


b)Β βˆ€xP(x)=P(-5)\land P(-3)\land P(-1)\land P(1)\land P(3)\land P(5)


c)

βˆ€π‘₯((π‘₯ β‰  1) β†’ 𝑃(π‘₯))=βˆ€π‘₯(\neg(x\neq 1)\lor P(x))=(x=1)\lor P(1)

d)

βˆƒπ‘₯((π‘₯ β‰₯ 0) ∧ 𝑃(π‘₯))=((x=1)\land P(1))\lor ((x=)\land P(3))\lor ((x=5)\land P(5))


e)

βˆƒπ‘₯(ᅭ𝑃(π‘₯)) ∧ βˆ€π‘₯((π‘₯ < 0) β†’ 𝑃(π‘₯)) =βˆƒπ‘₯(ᅭ𝑃(π‘₯)) ∧ βˆ€π‘₯(\neg(π‘₯ < 0) \lor 𝑃(π‘₯))=

=βˆƒπ‘₯(ᅭ𝑃(π‘₯)) ∧ βˆ€π‘₯((π‘₯ > 0) \lor 𝑃(π‘₯))=

=βˆƒπ‘₯(ᅭ𝑃(π‘₯)) ∧ (((π‘₯ =1) \lor 𝑃(1))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))=


=(ᅭ𝑃(-5)) ∧ (((π‘₯ =1) \lor 𝑃(1))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))\lor

\lor (ᅭ𝑃(-3)) ∧ (((π‘₯ =1) \lor 𝑃(π‘₯))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))\lor

\lor (ᅭ𝑃(-1)) ∧ (((π‘₯ =1) \lor 𝑃(π‘₯))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))\lor

\lor (ᅭ𝑃(1)) ∧ (((π‘₯ =1) \lor 𝑃(π‘₯))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))\lor

\lor (ᅭ𝑃(3)) ∧ (((π‘₯ =1) \lor 𝑃(π‘₯))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))\lor

\lor (ᅭ𝑃(5)) ∧ (((π‘₯ =1) \lor 𝑃(π‘₯))\land ((π‘₯ =3) \lor 𝑃(3))\land ((π‘₯ =5) \lor 𝑃(5)))

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Question ID: mtid-5-stid-8-sqid-846-qpid-731