Contrapositive statement

In summary: It has to be "if ab is even then a and b are not both not even". Which is just a fancy way of saying "if ab is even then at least one of a and b is even". But this is not the contrapositive of "if a and b are non-even numbers then ab is non-even". The contrapositive of "if a and b are non-even numbers then ab is non-even" is "if ab is even then at least one of a and b is even". So the contrapositive of "if a and b are non-negative numbers then ab is non-negative" is
  • #1
kingwinner
1,270
0
Fact: If a and b are non-negative numbers, then ab is non-negative.

What is the equivalent contrapositive statement of the above?
I think it is:
If ab<0, then at least one of a and b <0.
Am I right?

But this implication doesn't seem quite right to me...shouldn't the correct statement be...
If ab<0, then exactly one of a and b <0 ?

Could someone please explain?

Thank you!
 
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  • #2
"If ab<0, then at least one of a and b <0" is the contrapositive, and it is perfectly true. "If ab<0, then exactly one of a and b < 0" is also true, but it is not the contrapositive (it says more than the contrapositive does).
 
  • #3
Some things to consider:
1) The statement S
"If x then y"
has the following http://www.acm.org/crossroads/xrds10-3/gfx/img1.gif":
For x false, y false the statement "If x then y" is true (vacuous truth)
For x false, y true the statement "If x then y" is true (vacuous truth)
For x true, y false the statement "If x then y" is false
For x true, y true the statement "If x then y" is true

(Instead of "If x then y" we also write "x => y")


2) The negation of "x and y"
is "(not x) or (not y)".
---------------------------

Now, to your statement:
[tex](a,b \geq 0) \Rightarrow (ab \geq 0)[/tex]
or equivalently
[tex](a \geq 0 \text{ and } b \geq 0) \Rightarrow (ab \geq 0)[/tex]

The contrapositive is:
[tex]\text{not }(ab \geq 0) \Rightarrow \text{not }( a \geq 0 \text{ and } b \geq 0)[/tex]
or equivalently
(ab < 0) => (a<0 or b<0)


Cases:
We will check for every case of a,b whether the statement
(ab <0) => (a<0 or b<0) is true.
(Let's call this statement S as in the beginning.
The x corresponds to "ab<0" and y corresponds to "a<0 or b<0".)


Examine statement S: "If (ab<0) then (a<0 or b<0)"

Case 1) a<0, b<0:
(ab<0) is false, (a<0 or b<0) is true
From our truth table we can conclude that S is true.

Case 2) a<0, b=0:
(ab<0) is false, (a<0 or b<0) is true
From our truth table we can conclude that S is true.

Case 3) a<0, b>0:
(ab<0) is true, (a<0 or b<0) is true
From our truth table we can conclude that S is true.

Case 4) a=0, b<0:
(ab<0) is false, (a<0 or b<0) is true
From our truth table we can conclude that S is true.

Case 5) a=0, b=0:
(ab<0) is false, (a<0 or b<0) is false
From our truth table we can conclude that S is true.

Case 6) a=0, b>0:
(ab<0) is false, (a<0 or b<0) is false
From our truth table we can conclude that S is true.

Case 7) a>0, b<0:
(ab<0) is true, (a<0 or b<0) is true
From our truth table we can conclude that S is true.

Case 8) a>0, b=0:
(ab<0) is false, (a<0 or b<0) is false
From our truth table we can conclude that S is true.

Case 9) a>0, b>0:
(ab<0) is false, (a<0 or b<0) is false
From our truth table we can conclude that S is true.

As you can see the statement S is true for all cases of a,b.
 
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  • #4
kingwinner said:
Fact: If a and b are non-negative numbers, then ab is non-negative.

What is the equivalent contrapositive statement of the above?
I think it is:
If ab<0, then at least one of a and b <0.
Am I right?

But this implication doesn't seem quite right to me...shouldn't the correct statement be...
If ab<0, then exactly one of a and b <0 ?

Could someone please explain?

Thank you!

What you are stumbling on is the fact that the converse of the original statement happens to be true also and its contrapositive along with the one you're being asked to find together make the "exaclty one" part true. But the contrapositive of the implication you're given is "at least one," and no further.

--Elucidus
 
  • #5
Thanks for the helpful comments!
 
  • #6
kingwinner said:
Fact: If a and b are non-negative numbers, then ab is non-negative.

What is the equivalent contrapositive statement of the above?
I think it is:
If ab<0, then at least one of a and b <0.
Am I right?

But this implication doesn't seem quite right to me...shouldn't the correct statement be...
If ab<0, then exactly one of a and b <0 ?

Could someone please explain?

Thank you!

You are partially correct. It seems obvious to me that ab<0 (P) iff exactly one of (a,b) < 0 and exactly one of (a,b)>0 (Q). The contrapositive of this is ~P-->~Q, ie: not(ab<0) iff not(exactly one of (a, b) < 0 and exactly one of (a,b)>0).

I don't see where ab<0 if at least one of (a,b)<0 is true since it doesn't exclude the possibility of (a,b) having the same sign. It's obvious that (a,b) cannot have the same sign and neither (a,b) can be zero if ab<0.
 
Last edited:
  • #7
SW VandeCarr said:
I don't see where ab<0 if at least one of (a,b)<0 is true since it doesn't exclude the possibility of (a,b) having the same sign. It's obvious that (a,b) cannot have the same sign and neither (a,b) can be zero if ab<0.
Yes, but the (false) statement "ab < 0 if at least one of a and b is less than 0" was never mentioned. Remember that "P if Q" means "if Q, then P."
 
  • #8
Since "NOT non-negative" is "negative", the direct contrapositive to "If a and b are non-negative numbers then ab is non-negative" is "If ab is negative then it is not the case that a and b are non-negative numbers".

But "not P and Q" is "not P or not Q" so "NOT (a and b are non-negative)" is "either a is negative or b is negative".

So the contrapositive of "if a and b are non-negative numbers then ab is non-negative" is "if ab is negative then either a is negative or b is negative".

While it is true that a and b can't both be negative, that fact does NOT follow from the original statement. That requires more information.

For example, suppose statement were "if a and b are non-even numbers then ab is non-even". What is the contrapositive of that?
 

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. For example, the contrapositive of "if it is raining, then the ground is wet" would be "if the ground is not wet, then it is not raining."

Why is a contrapositive statement important?

A contrapositive statement can be useful in proving the validity of a conditional statement. If the contrapositive is true, then the original statement must also be true. It also allows for the identification of equivalent statements, which can be helpful in logic and mathematics.

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

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

Can a contrapositive statement be written in a different form?

Yes, a contrapositive statement can be written in different forms, such as using equivalent statements or using symbols in place of words. For example, the contrapositive of "if x is an even number, then x+1 is an odd number" can also be written as "if x+1 is not an odd number, then x is not an even number."

How is a contrapositive statement different from a converse statement?

A contrapositive statement is formed by switching and negating the hypothesis and conclusion of a conditional statement, while a converse statement is formed by switching the hypothesis and conclusion without negating them. This means that the truth value of a contrapositive statement is always the same as the original statement, while the truth value of a converse statement may be different.

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