UGC NET CSE | June 2019 | Part 1 | Question: 26

Question

If the proposition ‘Houses are not bricks’ is taken to be False then which of the following propositions can be True?

1. All houses are bricks
2. No house is brick
3. Some houses are bricks
4. Some houses are not bricks

Select the correct answer from the options given below:

• A. ii and iii
• B. i and iv
• C. ii only
• D. iii only

Answer 1

Let $x$ : An object

      $h(x)$: $x$ is a house  and

      $b(x)$: $x$ is a brick

Houses are not bricks $\equiv$ For all $x$ , if $x$ is a house then it is not a brick  $\equiv \forall x( h(x) \rightarrow \sim b(x)) \equiv \forall x (\sim h(x)\ \vee \sim b(x))$

This above proposition is considered false so will negate it.

$\sim( \forall x (\sim h(x)\ \vee \sim b(x))) \equiv \exists x ( h(x)\ \wedge b(x)) $

1. All houses are bricks $\equiv$ For all $x$ , if $x$ is a house then it is also a brick $\equiv \forall x ( h(x) \rightarrow b(x)) \equiv \forall x (\sim h(x)\ \vee b(x))$
2. No house is a brick $\equiv$ There does not exist an $x$ such that  $x$ is a house and it is a brick $\equiv \sim \exists x ( h(x) \wedge b(x)) \equiv \forall x (\sim h(x)\  \vee  \sim b(x) )$
3. Some houses are bricks$\equiv$ There exists some $x$ such that $x$ is a house and it is a brick $\equiv \exists x(h(x) \wedge  b(x))$
4. Some houses are not bricks$\equiv$ There exists some $x$ such that $x$ is a house and it is not a brick $\equiv \exists x(h(x) \wedge \sim b(x))$

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

Comments

Some houses are bricks is surely true  but All houses are bricks may be true or false