Translating sentences into Statements

[Concor]

Hello i am working through a math textbook in proofs.

i have gotten to a question that i dont understand why the answer is the way it is.

in the (a) and (b) question the answer does not negate the variable being integer whereas in the (c) both "r does not equal 0" and "r is a rational number" is negated. Why? is it because in the (A) and (b) questions the assignment to the integers is not part of the implication?
a) (a,b,c belong to the integers) and (if (a|b and b|c) then a|c)
but
(c) if (r belongs to rationals and r =/= 0) then 1/r is rational

below are the answers



Thanks for any help.

 

[R.M.]

Check your definitions of those words! The converse does not "negate" anything. It merely switches the "if" and "then" parts of the statement.
Example Statement: if aaaaa, then bbbbb
Converse of Example statement: if bbbbb, then aaaaa
Inverse of Example statement: if NOT aaaaa, then NOT bbbbb (note the ORDER stays the same)
Contrapositive of Example statement: if NOT bbbbb, then NOT aaaaa (converse, but negated)

Here's one I use with my students:
Statement: if x = 3, then x² = 9
Converse: if x² = 9, then x = 3 {note that this is false, since x could also be -3}
Inverse: if x [imath]\neq[/imath] 3, then x² [imath]\neq[/imath] 9
Contrapositive: if x² [imath]\neq[/imath] 9, then x [imath]\neq[/imath] 3

 

[Dr.Peterson]

There is some ambiguity in this sort of question. The given statements ((a) and (b) in particular) are not merely "if P then Q", but rather "In context C, if P then Q". They have chosen, reasonably, to take being an integer as part of the context rather than the condition, so that, for example, in taking the inverse, they negate P and Q, not C. (Of course, nothing is negated in stating the converse, and I don't think you were saying they should.)

I've seen this called a "partial converse", and so on, but I prefer this idea of context, as my brother and I discussed here.

 

[Agent Smith]

Check your definitions of those words! The converse does not "negate" anything. It merely switches the "if" and "then" parts of the statement.
Example Statement: if aaaaa, then bbbbb
Converse of Example statement: if bbbbb, then aaaaa
Inverse of Example statement: if NOT aaaaa, then NOT bbbbb (note the ORDER stays the same)
Contrapositive of Example statement: if NOT bbbbb, then NOT aaaaa (converse, but negated)

Here's one I use with my students:
Statement: if x = 3, then x² = 9
Converse: if x² = 9, then x = 3 {note that this is false, since x could also be -3}
Inverse: if x [imath]\neq[/imath] 3, then x² [imath]\neq[/imath] 9
Contrapositive: if x² [imath]\neq[/imath] 9, then x [imath]\neq[/imath] 3

Great answer.

 

[Agent Smith]

Hey, I just discovered this:

The inverse of the converse is the contrapositive