I know this may seem like a really unintellectual question however I was wondering what statements and proposed negations are. My teacher didn't really go over this and I can't seem to find it in my textbook or any article online that can fully explain this concept in depth.
Are statements just a set of facts that we are supposed to defunct using the proposed negation?
Here are examples and I'm supposed to determine whether the proposed negation is correct. If it is wrong, I have to write a correct version. Then determine whether the original statement or its negation is true.
Statement: For all real numbers x, y, if x^2 > y^2 then x > y. Proposed Negation: There exists real numbers x,y such that x^2 > y^2 but x <= y.
Statement: For all real numbers x, if x!= 0 then x has a multiplicative inverse. Proposed Negation: There exists a nonzero real number that does not have a multiplicative inverse.
Statement: There exist odd integers whose product is odd. Proposed Negation: The product of any two integers is odd.
I can see both statements however what do I do with it? Help would be appreciated. Thanks!
A statement is a sentence that is either true or false. The negation of a statement would be simply the "opposite" of the statement.
1) The statement says "for all $x, y$ if $x^2=y^2$ then $x>y$ for all $x,y$"
$\quad$This is the same as saying "there $does\; not$ exist $x, y$ such that if $x^2>y^2$ but $x \le y$"
$\quad$The negation of the statement would then be:
$\quad$ there $exists$ $x, y$ such that $x^2 > y^2$ but $x \le y$
Do you know how to proceed now?
2) The statement says "for all $x$, if $x \ne 0$ then $x$ has a multiplicative inverse"
$\quad$ This is the same as "there $does \; not$ exist $x$ such that if $x \ne 0$, then $x$ has no inverse"
$\quad$ The negation would be:
$\quad$ "there exists $x \ne 0$ but $x$ has no inverse"