GATE Overflow | General Aptitude | Test 2 | Question: 4

Question

Choose the set in which the combinations are logically equivalent.

1. All flowers are roses.
2. No rose is a flower.
3. No flower is a rose.
4. Some flowers are roses.
5. No rose is not a flower.
6. All roses are flowers.

• A. (i), (v)
• B. (ii), (iv)
• C. (iii), (iv)
• D. (v),  (vi)

Answer 1

Let $x$ : An object

      $r(x)$: $x$ is a rose  and

      $f(x)$: $x$ is a flower

1. All flowers are roses. $\equiv$ For all $x$ , if $x$ is a flower then it has to be a rose  $\equiv$ $\forall x$($ f(x) \rightarrow r(x))$$\equiv$  $\forall x$$(\sim f(x)\ V r(x))$
2. No rose is a flower.$\equiv$ There does not exist an $x$ such that  $x$ is a rose and it is a flower $\equiv$ $\sim \exists x ( r(x) \Lambda f(x))$ $\equiv$ $\forall x$$(\sim r(x)\  V  \sim f(x) )$
3. No flower is a rose.$\equiv$ There does not exist $x$ such that $x$ is a flower and it is a rose $\equiv$ $\sim \exists x ( f(x) \Lambda r(x))$ $\equiv$ $\forall x$$(\sim f(x)\  V \sim r(x) )$
4. Some flowers are roses$\equiv$ There exists some $x$ such that $x$ is a flower and it is a rose $\equiv$ $\exists x(f(x) \Lambda  r(x))$
5. No rose is not a flower.$\equiv$ There does not exist $x$ such that $x$ is a rose and it is not a flower $\equiv$$\sim \exists x( r(x) \Lambda \sim f(x))$ $\equiv$ $\forall x$$(\sim r(x)\ V f(x))$
6. All roses are flowers. $\equiv$$\forall x$( if $x$ is a rose then it is a flower)$\equiv$ $\forall x$$( r(x) \rightarrow f(x))$ $\equiv$ $\forall x$$(\sim r(x)\ V f(x))$$\equiv$

$\therefore$ Option $D.$ is the correct answer.

Comments

@Satbir

5 is saying all roses are flowers,i.e rose is a subset of flower.but 6 is saying No rose is not a flower.

there might be some part(flower) which are not belongs to rose but belongs to flowers.then it should be No rose may or may not be a flower.

I think option a should be correct.

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I think only D is logically consistent.

In D, according to (v) which says “no rose is not a flower” (meaning, there is no rose which is not a flower, or, every rose is a flower) the Venn diagram will be where the set of all roses is a subset of the set of all flowers. And this clearly is consistent with (vi).

In A, according to (i) the Venn diagram will be where the set of all flowers is a subset of the set of all roses, and this cannot be consistent with (v).

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@Satbir option a is contrapositive right then why its not correct?