Taking the contrapositive of a statement I have a proofs question in which im proving a statement by using the contrapositive. My understanding of this is that we take "If A then B" and infer it from "If not A then not B", but I am unsure of how to apply this to my statement. It goes similar to this,

if xy < 140 then x < 10 or y < 14.

My first thoughts is this would translate to

If x ≥ 140 then x ≥ 10 or y ≥ 14

I don't see how this would be correct because using x = 8 and y = 14, for example, this would not be greater than or equal to 140, so I don't know how to correctly take the contrapositive of this statement, help would be greatly appreciated.
 A: Your mistake is that "NOT (A or B)" is "(NOT A) and (NOT B)".
So the contrapositive of "if xy< 140 then x< 10 or y< 14" is "if NOT (x< 10 or y< 14) then NOT xy< 140" which is
"if $x\ge 10$ and $y\ge 14$ then $xy \ge 140$".
A: If $A$ is $xy<140$ and $B$ is $x<10$ or $y<14$, then proving “if $\neg B$ then $\neg A$” means to prove$$x\geqslant10\text{ and }y\geqslant14\implies xy\geqslant14,$$which is clearly true.
A: Hint: the statement is $P\Rightarrow (Q\vee R)$.
The contrapostive is $\neg (Q\vee R)\Rightarrow\neg P$ or with De Morgan, $(\neg Q\wedge \neg R)\Rightarrow \neg P$.
A: The contrapositive of "If A then B" isn't "If not A then not B" but "If not B then not A".  Both negating A and B and changing the direction of the implication are essential!  As you've found, if you only do one of those things, you won't end up with an equivalent statement.
So, for example, the contrapositive of

if xy < 140 then x < 10 or y < 14

is:

if not(x < 10 or y < 14) then not(xy < 140)

or, equivalently by De Morgan's law:

if not(x < 10) and not(y < 14) then not(xy < 140)

which we can write more compactly as:

if x ≥ 10 and y ≥ 14 then xy ≥ 140.

