# convers and contrapositive question

• Nov 16th 2013, 06:16 AM
ronanbrowne88
convers and contrapositive question
Could some one with more knowledge than me just confirm if i am doing the right thing here, Thanks.

State
(i) the converse, and
(ii) the contrapositive
of the following implication:

if a > b and b > c and a > c

(i) a > c only if a > b and b > c
(ii) if c > a then b > a and c > b
• Nov 16th 2013, 06:29 AM
HallsofIvy
Re: convers and contrapositive question
The terms "converse" and "contrapositive" apply only to statements of the form "If A Then B". In that case the "converse" is "If B Then A" and the "contrapositive" is "If NOT B then NOT A".

But you do NOT have an implication here because you have no "then".

You probably mean "if a> b an b> c then a> c". (Which is true.)

The "converse" is "if b> c then a>b a and b>c" (Which is not true.) The "only if" formulation is equivalent to the "if then" formulation but "if X then Y" is equivalent to "X only if Y". You have "X" and "Y" reversed.

The contrapositive is "if NOT (a> c) then NOT (a> b and b> c)". We can reduced that a little by using the fact that "NOT (X and Y)" is the same as "NOT X or NOT Y" as well as the fact that "NOT x> y" is the same as "\$\displaystyle x\le y\$" to write the contrapositive as "if \$\displaystyle a\le c\$ then either \$\displaystyle a\le b\$ or \$\displaystyle b\le c\$". (which is true. The contrapositive of a statement is true if and only if the statement is true.)

You seem to think that "NOT x> y" is the same as "x< y" (y> x) but that is not true. If x= y then "NOT x> y".
• Nov 16th 2013, 08:25 AM
ronanbrowne88
Re: convers and contrapositive question
Quote:

Originally Posted by HallsofIvy
The "converse" is "if b> c then a>b a and b>c" (Which is not true.) The "only if" formulation is equivalent to the "if then" formulation but "if X then Y" is equivalent to "X only if Y". You have "X" and "Y" reversed.

Thanks for the help the makes things clearer, i presume that extra a above is a typo , after a >b?