don't understand part of answer to contrapositive question

Find the contrapositive. If there exists real numbers x and y with x $\displaystyle \neq$ y and $\displaystyle x^{2}+x y+y^{2}+x+y=0$ then f is not one-to-one

I got:

if f is one-to-one then for all real numbers x and y with x=y or $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$

Why is it still = and not $\displaystyle \neq$?

Re: don't understand part of answer to contrapositive question

Hey Jskid

When thinking about the contrapositive consider this.

The original statement: if m then n

Then the contrapositive will be: if not n then not m.

does this help?

P.S. to avoid those nasty latex errors use "tex" tags not "math" tags.

Re: don't understand part of answer to contrapositive question

Quote:

Originally Posted by

**pickslides** Hey Jskid

When thinking about the contrapositive consider this.

The original statement: if m then n

Then the contrapositive will be: if not n then not m.

does this help?

P.S. to avoid those nasty latex errors use "tex" tags not "math" tags.

A little.

So n is "If there exists real numbers x and y with $\displaystyle x \neq y$ and $\displaystyle x^{2}+x y+y^{2}+x+y=0$" right?

Why does the negation change x=y to $\displaystyle x \neq y $ the "and" to "or" but not $\displaystyle x^{2}+x y+y^{2}+x+y = 0$ to $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$?

Re: don't understand part of answer to contrapositive question

We have $\displaystyle \neg(A\land B)\Leftrightarrow\neg A\lor\neg B\Leftrightarrow A\to\neg B$.

Quote:

Originally Posted by

**Jskid** Find the contrapositive. If there exists real numbers x and y with x $\displaystyle \neq$ y and $\displaystyle x^{2}+x y+y^{2}+x+y=0$ then f is not one-to-one

I got:

if f is one-to-one then for all real numbers x and y with x=y or $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$

Should be: "...then for all real numbers x and y, it is the case that x=y or $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$." (The phrase "it is the case that" is inserted just to avoid two mathematical expressions with no words between them.) Alternatively, "...then for all real x and y, if $\displaystyle x\ne y$, then $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$."

Quote:

Originally Posted by

**Jskid** Why is it still = and not $\displaystyle \neq$?

What is "still" = ?. The assumption was $\displaystyle x\ne y\land x^{2}+x y+y^{2}+x+y=0$; its negation is $\displaystyle x=y\lor x^{2}+x y+y^{2}+x+y\ne0$.

Quote:

Originally Posted by

**Jskid** So n is "If there exists real numbers x and y with $\displaystyle x \neq y$ and $\displaystyle x^{2}+x y+y^{2}+x+y=0$" right?

Yes, but without the initial "If."

Quote:

Originally Posted by

**Jskid** Why does the negation change x=y to $\displaystyle x \neq y $ the "and" to "or" but not $\displaystyle x^{2}+x y+y^{2}+x+y = 0$ to $\displaystyle x^{2}+x y+y^{2}+x+y \neq 0$?

The last equation does change to inequation.

Re: don't understand part of answer to contrapositive question

The answer key has

If f is one-to-one, then for all real numbers x and y either x=y or $\displaystyle x^2+xy+y^2+x+y=0$

is the end a typo? Shouldn't it be $\displaystyle \neq 0$?

Re: don't understand part of answer to contrapositive question