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Write the component statements of the following compound statements and check whether the compound statement is true or false:

i. 57 is divisible by 2 or 3.

ii. 24 is a multiple of 4 and 6.

iii. All living things have two eyes and two legs.

iv. 2 is an even number and a prime number.

0 2 Views | Posted 3 months ago
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    Answered by

    Payal Gupta | Contributor-Level 10

    3 months ago

    This is a short answer type question as classified in NCERT Exemplar

    ( i ) H e r e     t h e     g i v e n     s t a t e m e n t     i s     t h e     f o r m     p ∨ q     w h i c h     h a s     t h e     t r u t h     v a l u e     T     w h e n e v e r     e i t h e r           p     o r     q     b o t h     h a v e     t h e     t r u t h     v a l u e     T .         H e n c e ,     I t     i s     a     t r u e     s t a t e m e n t     a n d     i t s     c o m p o n e n t     s t a t e m e n t s     a r e         p : 5 7     i s     d i v i s i b l e     b y     2 .                                                               ( F a l s e )         q : 5 7     i s     d i v i s i b l e     b y     3 .                                                               ( T r u e ) ( i i ) H e r e     t h e     g i v e n     s t a t e m e n t     i s     t h e     f o r m     p ∧ q     w h i c h     h a s     t h e     t r u t h     v a l u e     T     w h e n e v e r     b o t h             p     o r     q     h a v e     t h e     t r u t h     v a l u e     T .           H e n c e ,     I t     i s     a     t r u e     s t a t e m e n t     a n d     i t s     c o m p o n e n t     s t a t e m e n t s     a r e         p : 2 4     i s     a     m u l t i p l e     o f     4 .                                                               ( T r u e )         q : 2 4     i s     a     m u l t i p l e     o f     6 .                                                               ( T r u e ) (iii)It  is  a  false  statement.  Since  p∧q  which  has  the  truth  value  F  whenever  either           p     o r     q     o r     b o t h     h a v e     t h e     t r u t h     v a l u e     F .     I t s     c o m p o n e n t     s t a t e m e n t s     a r e         p : A l l     l i v i n g     t h i n g s     h a v e     t w o     e y e s .                     ( F a l s e )         q : A l l     l i v i n g     t h i n g s &thins

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Let Δ , ∇ ∈ { ∧ , ∨ } be such that p ∇ q ⇒ ( ( p Δ q ) ∇ r ) is a tautology. Then ( p ∇ q ) Δ   r  is logically equivalent to:
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Raj Pandey

Case – I Δ ≡ ∇ ≡ ∧  

              ( p ∧ q ) → ( ( p ∧ q ) ∧ r )  

              it can be false if r is false,

              so not a tautology

              Case – II If Δ ≡ ∇ ≡ ∨  

              ( p ∨ q ) ?     → ( ( p ∨ q ) ∨ r ) ≡ tautology

              then ( p ∨ q ) ∨ r ≡ ( p Δ r ) ∨ q   

              Case – I

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Negation of the Boolean statement ( p ∨ q ) ⇒ ( ( ∼ r ) ∨ p )  is equivalent to
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pva ( ∼ r   v   p )

∼ ( p ∨ q ) ∨ ( ∼ r v p )

its negation as asked in question

( p ∨ q ) ∧ ( ∼ p ∧ r )

= ( p ∧ ∼ p ∧ r ) ∨ ( q ∧ r ∧ ∼ p )

= ( p ∧ r ∧ ∼ p )                   [ a s     p ∧ ∼ p     i s     f a l s e ]

The remainder when (2021)2023 is divided by 7 is:
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Raj Pandey

2 0 2 1 ≡ − 2 mod (7)

⇒ ( 2 0 2 1 ) 2 0 2 3 ≡ ( − 2 ) 2 0 2 3 m o d ( 7 )   …… (i)

Now,   ( − 2 ) 3 ≡ − 1 m o d ( 7 )  

⇒ ( − 2 ) 2 0 2 3 ≡ ( − 2 ) m o d ( 7 ) ≡ 5 m o d ( 7 )       ……. (ii)

(i) & (ii)

⇒ ( 2 0 2 1 ) 2 0 2 3 ≡ 5 m o d ( 7 )  

∴  Remainder = 5

Let p, q, r be three logical statements. Consider the compound statements

  S 1 : ( ( ∼ p ) ∨ q ) ∨ ( ( ∼ p ) ∨ r )     a n d  

  S 2 : p → ( q ∨ r )  

 Then, which of the following is NOT true?
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kindly consider the following Image 

 

The statement ( ∼ ( p ⇔ ∼ q ) ) ∧ q     i s :  
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Raj Pandey

∼ ( p ⇔ ∼ q ) ∧ q = ( p ⇔ ∼ q ) ∧ q     i s :

∴ ( ∼ ( P ⇔ ∼ Q ) ) ∧ q is equivalent to ( p ⇒ q ) ∧

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