Question 198625
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The antecedent is the part of the statement immediately following the 'if'

The consequent is the rest of the statement.

The converse is formed by swapping positions of the antecedent and the consequent.  Where "If *[tex \LARGE p] then *[tex \LARGE q]" is the statement, the converse is formed "If *[tex \LARGE q] then *[tex \LARGE p]"

The inverse is formed by negation of both the antecedent and the consequent. Where "If *[tex \LARGE p] then *[tex \LARGE q]" is the statement, the inverse is formed "If not *[tex \LARGE p] then not *[tex \LARGE q]"

The contrapositive is formed by both negating and swapping the position of the antecedent and the consequent.  "If *[tex \LARGE p] then *[tex \LARGE q]" is the statement, the contrapositive is formed "If not *[tex \LARGE q] then not *[tex \LARGE p]"  A conditional statement and its contrapositive ALWAYS have the same truth value.  Note that the inverse is the contrapositive of the converse and vice versa.  Therefore the converse of a statement and the inverse of a statement ALWAYS have the same truth value.

So, for your number a.  The antecedent is "I go to work" and the consequent is "I will get paid"

Statement:  If I go to work, then I will get paid

Converse:  If I will get paid, then I go to work

Inverse: If I do not go to work, then I will not get paid

Contrapositive:  If I will not get paid, then I do not go to work

b. Antecedent: "You do not mow the grass".  Consequent: "Your yard will not look good"

Statement: If you do not mow the grass, then your yard will not look good.

Converse: If your yard does not look good, then you do not mow the grass.

For the inverse and contrapositive, note that two negatives cancel each other.  So the negative of "you do not mow the grass" is "you do not not mow the grass" which is equivalent to "you mow the grass"

Inverse: If you mow the grass, then your yard will look good.

Contrapositive:  If your yard looks good, then you mow the grass.

Now, let's analyze the truth values of the above.  Let's assume that the statement given in b is true.  It certainly makes sense that mowing the grass is one required element for a good-looking yard, at least for that set of yards that have a lawn.

Let's examine the truth of the contrapositive:  Since we accept the fact that mowing the grass is a required element of a good looking yard, if the yard looks good, it follows logically that the grass must be mowed (at least).

But now let's examine the converse.  Just because the yard doesn't look good, doesn't necessarily mean that the grass isn't mowed.  The lawn could be very neatly trimmed, but if you had not watered in six weeks, you would have a very neatly trimmed brown lawn that isn't very good looking.

Same problem with the inverse -- mowing the grass is necessary but NOT sufficient in and of itself to have a good looking yard.

One more thing, if you ever see the phrase "if and only if" that means that the statement and its converse are equivalent, as in "A triangle has 3 equal sides if and only if it has 3 equal angles"

John
*[tex \LARGE e^{i\pi} + 1 = 0]
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