GATE CSE 2012 | Question: 13

Question

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

“Some real numbers are rational”

• A. $\exists x (\text{real}(x) \lor \text{rational}(x))$
• B. $\forall x (\text{real}(x) \to \text{rational}(x))$
• C. $\exists x (\text{real}(x) \wedge \text{rational}(x))$
• D. $\exists x (\text{rational}(x) \to \text{real}(x))$

Comments

very true @R.Pradeesh Reddy I'm having difficulty in FOL so far in pyq's but this was easy and a little confidence boost 

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https://youtu.be/s41PLmHGf08?list=PLIPZ2_p3RNHidhFvyKH4U_VE1rFAyvG8U&t=2637

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meaning of All option : https://gateoverflow.in/45/gate-cse-2012-question-13?show=537679#a537679

Answer 1

Meaning of each choices:

• A. There exists a number which is either real or rational
• B. If a number is real it is rational
• C. There exists a number which is real and rational
• D. There exists a number such that if it is rational, it is real

So, (C) is the answer.

Comments

@Gate Fever, option d says, If there exists a rational number, it is real. Which isn’t same as statement asked.

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AND (^) : both predicate is true in statement no matter what value of x
implication (-->) : use not necessarily left predicate is true always, but whenever it is true, then right predicate must also be true.

Now in your example : 
Friends ^ Perfect means : there exist friends and is perfect
Friends --> Perfect means : if friend exist, then it must be perfect.

 

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got it

Answer 2

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

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. :)

Answer 3

(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 4

short trick

Use ^ for     “there exist(∃)”

use → for     “for ALL”

according to this option c is correct

Answer 5

Ans is C

Answer 6

Option C is correct

A) There exists an entity $x$ in the universe that is either a real number OR a rational number (or both)

B ) All real numbers are rational

C) There exists an entity $x$ such that $x$ is a real number AND $x$ is a rational number  means Some real numbers are rational.

D) There exists an entity $x$ such that, if $x$ is a rational number, then $x$ is a real number.