Question

If p → q is true, then ~ p → ~ q is never
true.

Answer 1

[Note : refer the attachments for the truth table ]

$\boxed{\textbf{\large{Step by step explanation}}}$

◾In a mathematical logic,

( → ) stands for implication (conditional)

If p and q are any two simple statements, then the compound statement 'if p then q' , meaning" statement p implies q " or"statement q is implied by statement p " is called a conditional statement. denoted by ( p → q)

As if we consider, a truth table of the conditional

[ refer the attachment for the truth table ]

◼ truth table for implication says that conditional statement is false if and only if p statement is true and q statement is false

◼( ~ ) stands for negation of that statement , if p is a statement then negation of p i.e 'not p' , negation of any simple statement p can also be

written as ' it is not true that ' or

'it is false that

[truth table of negation is in the attachment ]

◾As we have given the condition

[If (p → q) is true, then (~ p → ~ q) is never true]

the above condition is false, because (~ p → ~ q) is known as the inverse of the statement (p → q )

[ refer the attachment for the truth table of( p → q ) and (~ p → ~ q) ]

As if we consider the truth values of (p → q )and( ~ p → ~ q )in the truth table the values of p → q and (~ p → ~ q ) are T (true) at the two times i.e if (p → q) is true then (~ p → ~ q) is also true,therefor it says that if truth value of (p → q )is true then truth value of (~ p → ~ q) is not always false.