Does the contrapositive statement require changing and to or?

[Gear300]

The statement is:
If α is one-to-one and β is onto, then βoα is one-to-one and onto.
One-to-one is injection, onto is surjection, and being both is bijection. After showing that the statement is false, the contrapositive was asked for. The answer given is:
If βoα is not one-to-one and onto, then α is not one-to-one or β is not onto.
They changed the "and" to an "or." I was thinking that the "and" would be conserved in the contrapositive statement. Is it valid or necessary to change an "and" to an "or" for contrapositive statements?

 

[yyat]

The opposite of "A and B" is "not A or not B". This is one of http://en.wikipedia.org/wiki/DeMorgan%27s_laws" . So yes, you need to change "and" to "or" in this case when forming the contrapositive.

 

[HallsofIvy]

The contrapositive of "if A then B" is "if not B then not A",

But your question is not really about the contrapositive, it is about "Not (A and B)".

"Not (A and B)" is the same as saying "(not A or (not B)".

This is because "A and B" is true only if A= T and B= T. If A= T, B= F; A= F, B= T; or A= F, B= F, "A and B" is false. "Not (aA and B)" must be true in exactly those cases. In particular, it must be true in the cases A= T, B= F and A= F, B= T. That is precisely "(not A) or (not B)".

"(Not A) and (Not B)" would be true only in the case A= F, B= F.

 

[matt grime]

No, the contrapositive is as stated: it is an or. You are negating things. The negation of

A and B

is

not A or not B

so it is both necessary and valid.

Think about it: suppose A and B together imply C

Then "not C" can only happen if at least one of A or B is not true, and that's not A or not B.

 

[Gear300]

If βoα is not one-to-one and onto...

So does that imply the above quote would equivalently be written as If βoα is not one-to-one or βoα is not onto...?

 

[yyat]

So does that imply the above quote would equivalently be written as If βoα is not one-to-one or βoα is not onto...?

Correct.

 

[Gear300]

Thanks