determine converse is true or false ..easy question

• May 10th 2014, 07:31 AM
nikconsult
determine converse is true or false ..easy question
Kindly refer to the attached file. I just to confirm the books ans.

The book Ans : False
My Ans : True

Kindly confirm
TQ

Sorry if i post on the wrong section
• May 10th 2014, 07:44 AM
Jester
Re: determine converse is true or false ..easy question
What if x = 6?
• May 10th 2014, 07:52 AM
nikconsult
Re: determine converse is true or false ..easy question
Quote:

Originally Posted by Jester
What if x = 6?

Ans will be wrong
• May 10th 2014, 07:55 AM
nikconsult
Re: determine converse is true or false ..easy question
it make me confuse.If i accept the book answer (false e.g x = 6) but when i subtitute x = 72, the answer will be true.

correct me if i'm wrong
• May 10th 2014, 08:02 AM
Jester
Re: determine converse is true or false ..easy question
The counter example shows the converse is not true.
• May 10th 2014, 08:02 AM
Plato
Re: determine converse is true or false ..easy question
Quote:

Originally Posted by nikconsult
it make me confuse

Do you know what converse means.

If $x$ is a multiple of twelve then $x$ is a multiple of three. TRUE.

CONVERSE: If $x$ is a multiple of three then $x$ is a multiple of twelve. FALSE. Example $x=15$.
• May 10th 2014, 08:05 AM
nikconsult
Re: determine converse is true or false ..easy question
dear ploto & jester

If i accept the book answer (false e.g x = 6 or 15) but when i subtitute x = 72, the answer will be true.

correct me if i'm wrong
• May 10th 2014, 08:12 AM
Plato
Re: determine converse is true or false ..easy question
Quote:

Originally Posted by nikconsult
dear ploto & jester
If i accept the book answer (false e.g x = 6 or 15) but when i subtitute x = 72, the answer will be true.
correct me if i'm wrong

The statement "If $x$ is a multiple of twelve then $x$ is a multiple of three."
is true if and only if it holds for every $x$

The statement "If $x$ is a multiple of three then $x$ is a multiple of twelve." does not hold for every $x$ so it is false.

• May 10th 2014, 08:18 AM
nikconsult
Re: determine converse is true or false ..easy question
Quote:

Originally Posted by Plato
The statement "If $x$ is a multiple of twelve then $x$ is a multiple of three."
is true if and only if it holds for every $x$

The statement "If $x$ is a multiple of three then $x$ is a multiple of twelve." does not hold for every $x$ so it is false.

Dear Mr Plato...
tq for your respond...it mean FALSE ans is correct??
• May 10th 2014, 10:37 AM
bkbowser
Re: determine converse is true or false ..easy question
$P \rightarrow Q$, which is read as "If P then Q", is equivalent to $(\neg P \lor Q)$, which is read as "either not P or Q". This statement is only false when the antecedent, P, is true while the consequent, Q, is false. Where P, Q are well formed formulas.

In this particular case you have taken the statement, If x is a multiple of 12, then x is a multiple of 3, and correctly converted it into its consequent, If x is a multiple of 3, then x is a multiple of 12. You now have to either prove the statement or find a counterexample (The question asks you to do this when it says "state whether the converse is true or false").

For example consider $x=9$. Substituting 9 for x in the converse of the original conditional statement we have: If 9 is a multiple of 3, then 9 is a multiple of 12.

Recall the definition of a multiple; In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if $b = na$ for some integer n. (This is from wikipedia)

Evaluate the antecedent first, or "9 is a multiple of 3". Since $9=3^2=3*3$ we conclude that 9 is indeed a multiple of 3.

Evaluating the consequent second, which is "9 is a multiple of 12". We have it, from the definition of multiple that there exists $n \in \mathbb{N}$ such that $12n=9$. Since $n=\frac{9}{12}=\frac{3}{4}$ n is a rational number, which contradicts our assumption $n \in \mathbb{N}$. Hence we are entitled to conclude the negation of the consequent.

So we have a situation where the antecedent is true, while the consequent is false. Hence the conditional is false.