#48

mmax

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If the given is not in if-then form, rewrite it. Write the converse, inverse, and contrapositive of each conditional statement. Determine the truth value of each statement.

If 3x-7=20, then x=9
 
Hello, mmax!

Do you know what the converse, inverse and contrapositive are?
If you do, then where is your difficulty?


If the given is not in if-then form, rewrite it.
Write the converse, inverse, and contrapositive of each conditional statement.
Determine the truth value of each statement.

If 3x-7 = 20, then x = 9

\(\displaystyle \begin{array}{ccccc} \text{Implication:} & \text{If }3x-7 \,=\,20\text{, then }x\,=\,9 \\ \text{Converse:} & \text{If }x \,=\,9\text{, then }3x-7 \,=\,20 \\ \text{Inverse:} & \text{If }3x-7\,\ne\,20\text{, then }x \,\ne\,9 \\ \text{C'positive:} & \text{If }x\,\ne\,9\text{, then }3x-7\,\ne\,20 \end{array}\)

All four statements are valid.
 
Hello, mmax!

Do you know what the converse, inverse and contrapositive are?
If you do, then where is your difficulty?



\(\displaystyle \begin{array}{ccccc} \text{Implication:} & \text{If }3x-7 \,=\,20\text{, then }x\,=\,9 \\ \text{Converse:} & \text{If }x \,=\,9\text{, then }3x-7 \,=\,20 \\ \text{Inverse:} & \text{If }3x-7\,\ne\,20\text{, then }x \,\ne\,9 \\ \text{C'positive:} & \text{If }x\,\ne\,9\text{, then }3x-7\,\ne\,20 \end{array}\)

All four statements are valid.

I feel compelled to point out that all four of the statements in Soroban's example are valid.

But that's not generally true!

The converse and the inverse of a VALID conditional statement may NOT be valid; however, the contrapositive of a valid conditional statement will be valid also.

Consider this conditional statement:

If x = 5, then x < 10.

That's valid, right?

Now, what about the converse. Remember that the converse of the statement "if p then q" is "if q then p."

So the converse of "if x = 5, then x < 10" is "If x < 10, then x = 5." That's not true!

The inverse of "if p then q" is "If not-p, then not-q". For our example conditional statement, the inverse would be "If x is not equal to 5, then x is not less than 10". Is that true? I don't think so! What if x is 0?

The contrapositive of the statement "if p then q" is "if not-q, then not-p."

For the example, the contrapositive would be "if x is NOT less than 10, then x is not equal to 5." Pick any number that is not less than 10 (must be greater than or equal to 10, then)...can you feel quite sure that the number you pick is certainly NOT equal to 5? I think so.

To summarize:

statement: If x = 5, then x < 10.
converse: If x < 10, then x = 5.
inverse: If x is not equal to 5, then x is not less than 10.
contrapositive: If x is not less than 10, then x is not equal to 5.

The initial statement is valid, but of the other three, only the contrapositive is valid.

For a conditional statement if p then q, only the contrapositive if not-q then not-p is guaranteed to have the same truth value (be logically equivalent).
 
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