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Discrete Mathematics

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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, ints inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false.

  • Discrete Mathematics -

    Here's a summary drawn from:
    http://www.jimloy.com/logic/converse.htm

    statement: if p then q
    converse: if q then p
    inverse: if not p then not q
    contrapositive: if not q then not p

    For the given example:
    Statement:
    ∀x∈R, if(x+2)(x-3) > 0 then x<-2 or x>3

    So
    p: if (x+2)(x-3)
    q: x<-2 ∨ x>3

    Substitute into the above to find the inverse, converse and contrapositive. Post your answers for checking if you wish.

  • Discrete Mathematics -

    correction:
    p: (x+2)(x-3) >0
    q: x<-2 ∨ x>3

  • Discrete Mathematics -

    I don't quite understand what your trying to say here. Is this suppose to be an Inequality or Algebraic Equation or what is it that I am actually suppose to solve?

  • Discrete Mathematics -

    The example is to apply mathematical logic to the results of the solution of an algebraic inequality.
    There is nothing to do algebraically, but to modify the original statement to make the converse, inverse and contrapositive.

    If the original statement were:
    "If it rains, then I go to the park".
    Compare with the symbolic logic statement:
    if p then q ( p -> q )
    we conclude that
    p = it rains (condition)
    q = I stay home (consequence).

    The converse is then
    if q then p (q -> p)
    which translated in words:
    if I stay home then it rains.

    The inverse is:
    if ~p then ~q (~p -> ~q)
    which translates to:
    If it does not rain, then I do not stay home.

    The contrapositive is:
    if ~q then ~p (~q -> ~p)
    which translates to:
    If I do not stay home, then it does not rain.

    So for the given question, you only need to substitute
    p = (x+2)(x-3) > 0
    q = x<-2 ∨ x>3
    and repeat the above exercise.

  • Discrete Mathematics -

    Let me see if I have this right and what your saying:

    (x + 2) (x - 3) > 0

    x squared - 3x + 2x -6 > 0

    x squared - 1x - 6 > 0

    Is this how you solve the problem?

  • Discrete Mathematics -

    I don't see any numerical value of what p and q are equal to so I am only understanding that the problem in front of me and what I am looking at shows me an inequality. What is it that I am missing here? I don't quite understand exactly what it is that I am suppose to insert or substitute.

  • Discrete Mathematics -

    There is no algebraic manipulation to be done, just like the raining and staying home.

    Let
    p = (x+2)(x-3) > 0
    q = x<-2 ∨ x>3

    Then
    p->q, or "if p then q", would translate to the original statement of

    If (x+2)(x-3)>0 then x<-2 ∨ x>3.

    The converse is then
    q->p, or "if q then p" translates to
    If x<-2 ∨ x>3 then (x+2)(x-3)>0

    There is no algebraic calculation to be done. You would substitute
    (x+2)(x-3)>0 for p, and
    x<-2 ∨ x>3 for q.

    I will let you do the inverse (~p->~q) and the contrapositive (~q->~p).

  • Discrete Mathematics -

    I am not sure what this one is but I will see whether or not I get this right.

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

  • Discrete Mathematics -

    Also, what does the wavy line mean when you mentioned the Inverse:

    If ~ p then ~ q (~ p - > ~ q)

    Another question what does the dash mean after the p in the parentheses? Is that supposed to be a dash, hyphen, minus, less than or equal to, or greater than or equal to?

  • Discrete Mathematics -

    Another question also is the example where you did the converse is that true or false?

  • Discrete Mathematics -

    "I am not sure what this one is but I will see whether or not I get this right.

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

    "

    This translates to If q then p, which is the converse of the original statement if p then q.

    Remember at the beginning I wrote:
    statement: if p then q
    converse: if q then p
    inverse: if not p then not q
    contrapositive: if not q then not p


    The wavy line means negation.
    If p means "it rains", then ~p means "it does not rain".

    You can (and should) read up about negation at :
    http://www.mathgoodies.com/lessons/vol9/negation.html

    Here's one of many sites which explain mathematical logic, in case you were absent when your teacher explained it at school.
    http://www.mathgoodies.com/lessons/toc_vol9.html

    -> is just a lazy way to write the right arrow, or &rarrow;. It is not meant to be a hyphen followed by a greater than sign.

    The original statement:
    "If it rains then I stay home."
    and its converse
    "If I stay home then it rains"
    are not equivalent statements.
    So even if we know the truth value of the original statement, we do not know the truth value of the converse.

    However, the original statement and the contrapositive are equivalent statements, which means that if the former is true, so is the latter.

    I suggest you read about mathematical logic, especially on the symbols, before you attempt any of the problems.

  • Discrete Mathematics -

    Would one of these be the 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

  • Discrete Mathematics -

    No, let me try this again would the Inverse be one of the following:

    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

  • Discrete Mathematics -

    The inverse of
    If p then q
    would be
    If ~p then ~q (i.e. if not p then not q)

    We have
    p=(x+2)(x-3)>0,
    so
    ~p would be (x+2)(x-3)<0

    and since
    q=x<-2 ∨ x>3
    ~q would be x>-2 ∧ x<3 (de Morgan's law).

    If there is no typo, then we do not find the inverse in any of the four given cases.

  • Discrete Mathematics -

    When it comes to the Inverse then is the statement true or false?

  • Discrete Mathematics -

    If I make the statement:
    If it rains then I stay home.
    I did not say what happens if it does not rain, so we don't get to know.

    The inverse would be:
    If it does not rain then I do not stay home.
    Here it is making a statement which is not derived from the original statement. So we don't get to know if it is true or not.

    So in conclusion, if the original statement is true, the inverse is NOT automatically true. It may be false, but then it may be true. We don't get to know.

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