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In a domain of natural numbers, use $P(x)$ for $x$ is prime and $Q(x)$ for $x$ is even. Also the symbols $ \ <,>$ for $x$ and $=$ for $y$ can be used.

Then write the following sentences into quantified statements.

(i) Some primes are even.

(ii) All even numbers are greater than $ 1 $ .

Answer:

I tried to write as follows:

(i) 'some primes are even' can be written as $ \exists <,> (\wedge Q(x)) $

Am I right ? Any help ?

Michael Hardy
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MAS
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  • The proper way to include text within MathJax is by using \text{}, thus: $$ \begin{align} A & = B + C \ & = D+E, \text{by Xmith's theorem.} \end{align} $$ But in this case there was no reason for it to be within MathJax, so I edited accordingly. $\qquad$ – Michael Hardy Jul 18 '17 at 04:08
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    "Also the symbols $<,>$ for $x$ and $=$ for $y$ can be used." This sentence seems corrupted. Could you check your source? – Fabio Somenzi Jul 18 '17 at 04:12
  • We can replace $ x \ \ by \ \ the \ symbol \ <,> $ and $ \ y \ by \ the \ symbol \ = $ . That's it. – MAS Jul 18 '17 at 04:12
  • Can I write the first sentence as : $ ( \exists x) P(x) \rightarrow Q(x) $ – MAS Jul 18 '17 at 04:16
  • Yes you can, although, $~\exists x~(P(x)\to Q(x))$ would be clearer, but also incorect. $~$ What you want to say is: "There exists something which is prime and is even." Try again. – Graham Kemp Jul 18 '17 at 04:17
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    But it wouldn't be right... (I'm not talking of the style of parenthesization.) – Fabio Somenzi Jul 18 '17 at 04:17
  • $ \exists x(P(x) \rightarrow P(x) \wedge Q(x) )$ . Is it ? – MAS Jul 18 '17 at 04:21
  • I need the quantified statement only . – MAS Jul 18 '17 at 04:22
  • @mabmath. Universal quantifiers are restricted by implication; Existentials are restricted by conjunction. Don't missmatch them. – Graham Kemp Jul 18 '17 at 04:23
  • So how do I write ? – MAS Jul 18 '17 at 04:24
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    An English sentence of the form, "Some As are Bs" is first re-written as "There is an $x$ that is both an A and a B." From there the translation into formula is straightforward. – Fabio Somenzi Jul 18 '17 at 04:28
  • can I write the first statement as : $ (\exists x) (\exists y) [P(x) \wedge Q(y)] $. How can I write for the second sentence ? – MAS Jul 18 '17 at 04:29
  • You could, but it would not say what you need to say. Read it back into English: There's an $x$ that is a prime and there's a $y$ that is even. Not quite right... – Fabio Somenzi Jul 18 '17 at 04:30
  • That is ok, but How can I write the statement " all even numbers are greater than 1 " ? If I write as : $ (\forall x)(Q(x) >1) $. Is it partly ok ? – MAS Jul 18 '17 at 04:33
  • Partly OK, but $Q(x)$ is either true or false and cannot be compared to an integer. Instead, you can use $x > 1$ in your sentence. – Fabio Somenzi Jul 18 '17 at 04:35
  • But $ Q(x) $ means even numbers . – MAS Jul 18 '17 at 04:36
  • You want "for any number, if it is even, then it is greater than one." – Graham Kemp Jul 18 '17 at 04:36
  • please help me writing this one . Because I am in confusion . – MAS Jul 18 '17 at 04:38
  • There's a lot of detailed advice in the comments by Graham and me. Try to relax and follow it instead of rushing to the next guess. Of course, a number of people on this site could just provide you with the solution, but what would you learn? – Fabio Somenzi Jul 18 '17 at 04:40
  • "Some primes are even" must be translated as "there are some $x$ that are prime and even": $\exists x \ (P(x) \land Q/(x))$. – Mauro ALLEGRANZA Jul 18 '17 at 06:54
  • "All even numbers are greater than 1" must be transalted as: "for every number $x$, if $x$ is even, then $x$ is greater that 1": $\forall x \ (Q(x) \to (x > 1))$. – Mauro ALLEGRANZA Jul 18 '17 at 06:56
  • The issue is: it seems thta you have no symbol for $1$ in your language. – Mauro ALLEGRANZA Jul 18 '17 at 06:57

1 Answers1

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Existential quantification is restricted by conjunction.

"Some A-things are B-things," means "There exists something which is an A-thing and is a B-thing." $$\exists x~\big(A(x)\wedge B(x)\big)$$

Universal quantification is restricted by implication (conditional).

"All A-things are B-things," means "Take anything, if it is an A-thing, then it is a B-thing." $$\forall x~\big(A(x)\to B(x)\big)$$

Graham Kemp
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  • For the next one, I wrote as : $ (\forall x >1 ) (Q(x) >1 )$. is it ok ? – MAS Jul 18 '17 at 04:56
  • No. $(Q(x)>1)$ would be "the truth-value of '$x$ is even' is greater than 1." You cannot compare a predicate to an integer. – Graham Kemp Jul 18 '17 at 05:14
  • $ (\forall x) [(x>1) \rightarrow greater (Q(x), 1)]$. This one is valid or not – MAS Jul 18 '17 at 05:20