Let p be the statement “x is an irrational number,'' q be the statement ''y is a trascendental number'', and r be the statement ''x is a rational number iff y is a transcendental number ''. Statement-1 r is equivalent to either q or p. Statement-2:r is equivalent to `(p harr~q)`

Options

A. Statement-1 is True, Statement -2 is Ture, Statement -2 is a correct explanation for statement -1
B. Statement-1 is True, Statement -2 is True,, Statement -2 is not a correct explanation for statement -1
C. Statement -1 is True , Statement -2 is false.
D. Statement -1 is False, statement -2 is True.

Answer

C

Explanation

`r= (~ p harrq)`
` Rightarrow r= (pvvq)^^(~q vv ~ p) [ because p harr q = (~ pvv q) ^^ (~ q vv p)]`
`Rightarrow r={( p vvq) ^^~q}vv {(pvvq)^^ ~ p}`
`Rightarrowr={(pvvq) ^^ ~q}vv{(pvvq) ^^ ~ p}`
`Rightarrow r= {(p ^^~q)vvc}vv( c vv ( q ^^ ~ p}`
`Rightarrow ( p ^^ ~q) vv ( q ^^ ~p)`
` Rightarrow ` r= either q or p
So, statment -1 is true.
Statement -2 is
` ~ ( p harr ~ q)`
`{(~p vv~q)^^ ( p vv q)}[because p harr q = (~ p vv q) ^^ ( ~ q vv p)]`
`= ~ {~( p vvq) ^^ ( p vvq)} = ( p ^^q)vv( ~ p ^^~q)`
Clearly, this is not equivalent to r.