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

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".

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?