Negation of "If ... then" statementsI am just being introduced to how logic is used in mathematics and my lecturer mentioned that ∼ ( P → Q ) ≡ p ∧ ∼ q. This is quite hard to grasp at first glance, so he gave an example: The negation of "if x ≠ 0 then y = 0" is " x ≠ 0 ∧ y ≠ 0".Well, my question is, why should that be the case? Why is the negation of "if x ≠ 0 then y = 0" not " x ≠ 0 ∧ y = 0"?Any explanations will be greatly appreciated :)

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Negation of &quot;If ... then&quot; statementsI am just being introduced to how logic is used in mathematics and my lecturer mentioned that   ∼  (  P  →  Q  )  ≡  p     ∧  ∼  q. This is quite hard to grasp at first glance, so he gave an example: The negation of &quot;if   x  ≠  0 then   y  =  0&quot; is &quot;  x  ≠  0  ∧  y  ≠  0&quot;.Well, my question is, why should that be the case? Why is the negation of &quot;if   x  ≠  0 then   y  =  0&quot; not &quot;  x  ≠  0  ∧  y  =  0&quot;?Any explanations will be greatly appreciated :)

sengsemmxa · Accepted Answer

One can show   A  ⇒  B  ≡  ¬  A  ∨  B using truth tables. By De Morgan&#039;s laws one concludes  x  ≠  0     ∧     y  =  0 does not negate the initial statement, but implies it, in fact. For if &quot;  x  ≠  0     ∧     y  =  0&quot;, then certainly &quot;if   x  ≠  0, then   y  =  0&quot;.

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