Negation of "If ... then" statements I 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!= then " 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"?

umemezelenqp

umemezelenqp

Answered question

2022-12-17

Negation of "If ... then" statements
I 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 :)

Answer & Explanation

sengsemmxa

sengsemmxa

Beginner2022-12-18Added 7 answers

One can show A B ¬ A B using truth tables. By De Morgan's laws one concludes
x 0     y = 0 does not negate the initial statement, but implies it, in fact. For if " x 0     y = 0", then certainly "if x 0, then y = 0".

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