Question

Q: Express the following as existence assertions;
(a) The equation x3 = 27 has a natural number solution.
(b) 1,000,000 is not the largest natural number.
(c) The natural number n is not a prime.

Answer 1

A:
(a) \[\exists x(x\in\mathbb N)(x^3=27)\]

Answer 2

(b) \[\exists x(x\in \mathbb N)(x>10^6)\]

Answer 3

(a) i would write
\[ \large (\exists x\in\mathbb{N})(x^3=27) \]
or
\[ \large (\exists x)(x\in\mathbb{N}\wedge x^3=27) \]

Answer 4

(b)
\[ \large (\exists x\in\mathbb{N})(x>10^6) \]

(c)
\[ \large (\exists p\in\mathbb{N})(p\neq1\wedge p\neq n\wedge p\mid n) \]

Answer 5

(c) \[(\exists n\in \mathbb N)((\exists p,q\in\mathbb N)(pq=n) \wedge p,q\neq 1 )\]

Answer 6

i can't quite understand your answer for (c) @helder_edwin

Answer 7

for n not to be prime, it has to have a divisor different from 1 and n.

Answer 8

does my answer for (c) work /?

Answer 9

no.

Answer 10

n=6 p=2 q=3

Answer 11

6 is not prime,

Answer 12

but it satisfies what u wrote

Answer 13

isn't that what we wanted?

Answer 14

there is a natural number for which there are two other natural numbers none of which is one and such their product is the first number.

Answer 15

sorry. u r right. but i would write
\[ \large (\exists p,q\in\mathbb{N})(pq=n\wedge p,q\neq1) \]

Answer 16

thank you @helder_edwin

Answer 17

u r welcome

Answer 18

so these are my final answers A:
(a) \[(\exists x\in\mathbb N)(x^3=27)\]
(b) \[(\exists x\in \mathbb N)(x>10^6)\]
(c) \[(\exists p,q\in\mathbb N)((pq=n) \wedge (p,q\neq 1 ))\]

Answer 19

by the way. when u wrote
\[ \large \exists x(x\in\mathbb{N})\color{red}{(x^3=27)} \]
the red part now lies outside the range of the quantifier.

there can be only one set of parenthesis after the quantifier.
\[ \large (\forall x)(P(x))\color{red}{\text{whatever lies beyond this point is not linked with }\forall} \]
\[ \large (\exists x)(P(x))\color{red}{\text{whatever lies beyond this point is not linked with }\exists} \]

Answer 20

yes. those would be the final answers.

Answer 21

ok i think i understand

Answer 22

great.