Does the contrapositive statement require changing and to or?

In summary, the statement is that if α is one-to-one and β is onto, then βoα is also one-to-one and onto. However, when looking at the contrapositive, the "and" in the original statement needs to be changed to an "or" in order to maintain the truth value. This is due to the fact that the negation of "A and B" is "not A or not B". Therefore, the contrapositive is "If βoα is not one-to-one or βoα is not onto, then α is not one-to-one or β is not onto."
  • #1
Gear300
1,213
9
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?
 
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  • #2
The opposite of "A and B" is "not A or not B". This is one of http://en.wikipedia.org/wiki/DeMorgan%27s_laws" [Broken]. So yes, you need to change "and" to "or" in this case when forming the contrapositive.
 
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  • #3
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.
 
  • #4
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.
 
  • #5
Gear300 said:
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...?
 
  • #6
Gear300 said:
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.
 
  • #7
Thanks
 

What is a contrapositive statement?

A contrapositive statement is a logical statement that is formed by switching the hypothesis and conclusion of a conditional statement and negating both. It is a way of expressing the inverse of a conditional statement.

How is a contrapositive statement written?

A contrapositive statement is written in the form "If not Q, then not P". This is the inverse of the original conditional statement "If P, then Q". The negation of both the hypothesis and conclusion is what distinguishes a contrapositive statement from a converse statement.

What is the purpose of a contrapositive statement?

The purpose of a contrapositive statement is to show the logical equivalence of a conditional statement and its contrapositive. This means that if the original statement is true, then the contrapositive is also true. It can also be used to prove the validity of an argument or to disprove a statement by showing that its contrapositive is false.

How do you determine the truth value of a contrapositive statement?

The truth value of a contrapositive statement can be determined by evaluating the truth value of the original conditional statement. If the original statement is true, then the contrapositive will also be true. If the original statement is false, then the contrapositive will also be false.

Can a contrapositive statement be equivalent to a conditional statement?

Yes, a contrapositive statement and a conditional statement can be equivalent. This means that they have the same truth value under all circumstances. In other words, if the original statement is true, then so is the contrapositive and if the original statement is false, then so is the contrapositive.

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