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27 votes
27 votes

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational”

  1. $\exists x (\text{real}(x) \lor \text{rational}(x))$
  2. $\forall x (\text{real}(x) \to \text{rational}(x))$
  3. $\exists x (\text{real}(x) \wedge \text{rational}(x))$
  4. $\exists x (\text{rational}(x) \to \text{real}(x))$
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2 Comments

just a Hint Use ^ for there exist(∃)
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for ALL

∀x(rational(x)→real(x))
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3 Answers

43 votes
43 votes
Best answer

Meaning of each choices:

  1. There exists a number which is either real or rational
  2. If a number is real it is rational
  3. There exists a number which is real and rational
  4. There exists a number such that if it is rational, it is real


So, (C) is the answer.

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4 Comments

what is the problem in option d,

it also conveys the same meaning i guess;

there exists a number such that if it is rational then real.

and question also says that some real numbers are rational numbers.

where am i wrong??
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Hi @Ayush Upadhyaya sir,

 

I get confused when to translate a statement as AND (^) and when to use implication (-->) .

eg. None of my friends are  perfect... here should i consider Friends ^ Perfect  or Friends-->Perfect

 

For statement : Good mobile phones are not cheap.  (from gate 2014) i used implication and that leads to the right answer. 

 

Similarly Not all that glitters is gold(2014) also implication gives the correct result

Any tips or any source to refer pls

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suppose the number is complex so real(x) is false but real(x) → rational(x) will be true but rational numbers are not complex
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5 votes
5 votes

“ There exists a number which is real and rational” and this is equal to “Some real number are rational number.”

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4 Comments

Rational numbers are always real

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If possible, can you please draw the ven diagrams for other options given in question ? It might be helpful. @LakshmanPatelRJIT

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You can try it out.
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Please, check and correct if wrong.

https://gateoverflow.in/?qa=blob&qa_blobid=4820487837657352897

And which tool you used for image ? Don’t know why my image is rotated. :)

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4 votes
4 votes
(C) is the answer.

Translation of (C):" There exists a number which is real and rational " and this is eqt to “Some real numbers are rational”.
Answer:

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