Welcome to our community

Be a part of something great, join today!

Write inverse, converse, and contrapositive following statement

Joystar1977

Active member
Jul 24, 2013
119
Write the inverse, converse, and contrapositive of the following statement:

upside down A x E R, if (x + 2) (x - 3) > 0, then x < -2 or x > 3

Indicate which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a conterexample for each that is false.

Please help me with this math problem because I am totally lost and don't understand it at all.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
My guess is that you don't know the definitions of these types of statements. Why don't you start by reading about them in your textbook or Wikipedia? Note that in constructing the inverse, converse, and contrapositive you are supposed to leave the universal quantifier alone and just change the implication. For example, the converse of $\forall x\,(P(x)\to Q(x))$ is $\forall x\,(Q(x)\to P(x))$.

In plain text, you can write "for all" for ∀ and "in" for ∈.
 

Joystar1977

Active member
Jul 24, 2013
119
Doesn't the converse, contrapositive, and Inverse as follows:

q arrow p is the converse of p arrow q

slash bar q arrow slash bar p is the contrapositive of p arrow q

slash bar p arrow slash bar q is the inverse of p arrow q
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Yes, this is correct. But you need to write inverse, etc., for the concrete statement given in post #1.

I suggest using notations from this post.
 

Joystar1977

Active member
Jul 24, 2013
119
Is another way of saying this as follows:

Statement: if p then q

Converse: if q then p

Inverse: if not p then not q

Contrapositive: if not q then not p

Statement:

upside down A x E R, if (x + 2) (x -3) > 0, then x < -2 or x > 3

So

p : if (x + 2) (x - 3) > 0

q: x < -2 V x > 3

Am I suppose to solve this problem like an inequality or algebraic equation?
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Statement:

upside down A x E R, if (x + 2) (x -3) > 0, then x < -2 or x > 3

So

p : if (x + 2) (x - 3) > 0
No "if".

q: x < -2 V x > 3
Correct.

Am I suppose to solve this problem like an inequality or algebraic equation?
No, you are supposed to
Write the inverse, converse, and contrapositive of the following statement:

upside down A x E R, if (x + 2) (x - 3) > 0, then x < -2 or x > 3
Replace p and q in the statements from the beginning of post #5 by the expressions you found later in that post.

And please use the notation suggestion from post #2:
In plain text, you can write "for all" for ∀ and "in" for ∈.
 

Joystar1977

Active member
Jul 24, 2013
119
Is this right for the Converse?

q -> p, or "if q then p" translates to

If x < -2 V x > 3 then (x + 2) (x - 3) > 0

Would one of these be the contrapositive or inverse?

If x > -2 V x < 3 then (x +2) (x-3) < 0

If x > -2 V x < 3 then (x + 2) (x-3) > 0

If x > -2 V x > 3 then (x + 2) (x - 3) > 0

If x > -2 V x > 3 then ( x + 2) (x - 3) < 0
I don't quite understand this because what I see in front of my face is an Inequality
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Is this right for the Converse?

q -> p, or "if q then p" translates to

If x < -2 V x > 3 then (x + 2) (x - 3) > 0.
Yes, this is the converse.

Would one of these be the contrapositive or inverse?

If x > -2 V x < 3 then (x +2) (x-3) < 0

If x > -2 V x < 3 then (x + 2) (x-3) > 0

If x > -2 V x > 3 then (x + 2) (x - 3) > 0

If x > -2 V x > 3 then ( x + 2) (x - 3) < 0
No. As a first try, you could keep the negations. As I said, if you replace p with (x + 2)(x - 3) > 0 and q with x < -2 ∨ x > 3 in ~q -> ~p, you get ~(x < -2 ∨ x > 3) -> ~((x + 2)(x - 3) > 0). Simple, isn't it? Then we can apply simplification and remove negations if we want. Note that ~(x < y) is not (x > y), but (x ≥ y). To figure out ~(x < -2 ∨ x > 3) we apply the De Morgan's law: ~(A ∨ B) = ~A ∧ ~B. Therefore, ~(x < -2 ∨ x > 3) is ~(x < -2) ∧ ~(x > 3), which is x ≥ -2 ∧ x ≤ 3. Some people abbreviate this as -2 ≤ x ≤ 3. Altogether, the contrapositive is (x ≥ -2 ∧ x ≤ 3) -> (x + 2)(x - 3) ≤ 0.