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Find the component statements of the following compound statements:

i. Number 7 is prime and odd.

ii. Chennai is in India and is the capital of Tamil Nadu.

iii. The number 100 is divisible by 3, 11, and 5.

iv. Chandigarh is the capital of Haryana and U.P.

v. √ 7 is a rational number or an irrational number.

vi. 0 is less than every positive integer and every negative integer.

vii. Plants use sunlight, water, and carbon dioxide for photosynthesis.

viii. Two lines in a plane either intersect at one point or they are parallel.

ix. A rectangle is a quadrilateral or a 5-sided polygon.

0 2 Views | Posted 3 months ago
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  • 1 Answer

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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 )         p : N u m b e r     7     i s     p r i m e .          q:Number  7  is  odd. ( i i )       p : C h e n n a i     i s     i n     I n d i a .                     q : C h e n n a i     i s     t h e     c a p i t a l     o f     T a m i l     N a d u . ( i i i )     p : 1 0 0     i s     d i v i s i b l e     b y     3 .           q:100  is  divisible  by  11.           r:100  is  divisible  by  5. ( i v )     p : C h a n d i g a r h     i s     t h e     c a p i t a l     o f     H a r y a n a .                     q : C h a n d i g a r h     i s     t h e     c a p i t a l     o f     U . P . ( v )         p : 7     i s     a     r a t i o n a l     n u m b e r .                     q : 7     i s     a n     i r r a t i o n a l     n u m b e r . ( v i )     p : 0     i s     l e s s     t h a n     e v e r y     p o s i t i v e     i n t e g e r .                     q : 0     i s     l e s s     t h a n     e v e r y     n e g a t i v e     i n t e g e r . ( v i i )   p : P l a n t s     u s e     s u n l i g h t     f o r     p h o t o s y n t h e s i s .                     q : P l a n t s     u s e     w a t e r     f o r     p h o t o s y n t h e s i s .           r:Plants  use  carbon-dioxide  for  photosynthesis. ( v i i i ) p : T w o     l i n e s     i n     a     p l a n e     i n t e r s e c t     a t     o n e     p o i n t .                         q : T w o     l i n e s     i n     a     p l a n e     a r e     p a r a l l e l . ( i x )     p : A     r e c t a n g l e     i s     a     q u a d r i l a t e r a l .           q:A  rectangle  is  a  5-sides  polygon.

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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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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   

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pva ( ∼ r   v   p )

∼ ( p ∨ q ) ∨ ( ∼ r v p )

its negation as asked in question

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= ( 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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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 )  

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(i) & (ii)

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

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

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kindly consider the following Image 

 

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

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

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