GATE Mechanical 2018 Set 1 | GA | Question: 10

Question

Consider the following three statements:

1. Some roses are red.
2. All red flowers fade quickly.
3. some roses fade quickly.

Which of the following statements can be logically inferred from the above statements?

• A. If (i) is true and (ii) is false, then (iii) is false.
• B. If (i) is true and (ii) is false, then (iii) is true.
• C. If (i)  and (ii) are true, then (iii) is true.
• D. If (i)  and (ii) are false, then (iii) is false.

Comments

C.

some roses are red and all red flowers fade quickly => some roses fade quickly

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please correct me where i am wrong .

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@Ray Tomlinson 

A. If (i) is true and (ii) is false, then (iii) is false.

this option then becomes:

if (some roses are red and some red flower do not fade quickly) then all roses do not fade quickly.

it's like $P \rightarrow Q$

where, $P = $ some roses are red and some red flower do not fade quickly

$Q = $ all roses do not fade quickly

so, to check if this statement is always true we will try to make it false in atleast one case.

see in this venn-diagram, $P$ is true but $Q$ is false. and we know that if $P$ is true and $Q$ is false then $P \rightarrow Q$ is false. so, this statement can be made false in atleast one case.

hence, option A is false.

Answer 1

Some roses are red
and
All red flowers fade quickly
implies
Some roses fade quickly.
So, if (i) and (ii) are TRUE, (iii) must be TRUE.

Correct Answer: C.

Comments

@Arjun sir why can't be answer d?

Answer 2

$\color{Blue}{\text{Blue}}$ color shaded area represents, Some roses fade quickly.

$(I\: \wedge \: II) \implies III$

$(T \wedge T) \implies T$

So, the correct answer is $(C).$

Comments

fir to last option bhi ssahe ho jayaga asa ma @lakshman patel RJIT

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no see loption B example for it

Answer 3

option c is correct

Answer 4

The correct inference is:

(C) If (i) and (ii) are true, then (iii) is true.

Explanation:

If statement (i) is true, then some roses are red. If statement (ii) is also true, then all red flowers fade quickly. Since some roses are red and all red flowers fade quickly, it logically follows that some roses must fade quickly, making (iii) true.

Why the Other Options Are Incorrect:

Option (A) is incorrect because if (ii) is false, it only means not all red flowers fade quickly, but some might, so (iii) cannot be definitively false.

Similarly, (B) is incorrect because if (ii) is false, we don't have enough information to conclude that some roses definitely fade quickly.

Option (D) is incorrect because even if (i) and (ii) are false, we cannot infer that no roses fade quickly; roses might fade for reasons unrelated to color.

Thus, the only valid inference is (C): If (i) and (ii) are true, then (iii) is true.