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Study Notes 021:
By William E. Steinman:
Geometry 14 Reasoning:
Here are some definitions and rules.
These are not necessarily obvious.
The converse of a statement is the
statement that is formed by interchanging the hypothesis and conclusion.
The converse of a true statement is
not necessarily true.
For example:
The converse of ‘perch are
fish’ is ‘fish are perch.’
The negative of a statement is the
denial of the statement.
For example:
The negative of the statement
‘all men are dumb’ is the statement ‘all men are not
dumb.’ Depends on who you ask.
The inverse of a statement is formed
by denying both the hypothesis and conclusion.
The inverse of a true statement is
not necessarily true.
For example:
The inverse of the statement
‘men are animals’ is the statement ‘those who are not men are
not animals.’
The contrapositive of a statement is
formed by interchanging the negative of the hypothesis with the negative of the
conclusion. So, the contrapositive is the inverse of the converse and the
converse of the inverse.
Huh?
For example:
‘All not fish are not
perch’ is the contrapositive of ‘all perch are fish.’
A statement is considered false if
one false instance of the statement exists.
The converse of a definition is
true.
The converse of a true statement
other than a definition is not necessarily true.
The inverse of a true statement is
not necessarily true.
The contrapositive of a true
statement is true and the contrapositive of a false statement is false.
Logically equivalent statements are
pairs of related statements that are either both true or both false.
Therefore, a statement and its
contrapositive are logically equivalent.
And, the converse and inverse of a
statement are logically equivalent.
For example:
Take this statement: A triangle is a
polygon.
The contrapositive is: A plane
figure that is not a polygon is not a triangle.
The converse is: A polygon is a
triangle.
The inverse is: A plane figure that
is not a triangle is not a polygon.
The first two statements are
logically equivalent and in this case, both true.
The last two statements are
logically equivalent and in this case both false.
Lord preserve us!
We have more of this stuff:
A partial conversion of a theorem is
formed by interchanging any one condition of the hypothesis with one
consequence in the conclusion.
A partial inversion of a theorem is
formed by denying one condition in the hypothesis and one consequence in the
conclusion.
For example:
Given from study notes 10 that
supplementary angles are two angles whose total angle is 180o.
Then equal supplementary angles are
right angles.
Good enough!
Then also from SN10 the converses
become:
Congruent right angles are
supplementary.
Supplementary right angles are
congruent.
The inverses become:
Congruent angles that are not
supplementary are not right angles.
Supplementary angles that are not
congruent are not right angles.
Had enough?
Okay.
Let’s look at some more
principles.
If a statement and its converse are
both true, then he conditions in the hypotheses of the statement are necessary
and sufficient for its conclusion.
If a statement is true and its
converse is false, then the conditions in the hypotheses of the statement are
sufficient but not necessary for its conclusion.
If a statement is false and its
converse is true, then the conditions in its hypotheses are necessary but not
sufficient for its conclusion.
If a statement and its converse are
both false, then the conditions in the hypotheses are neither necessary nor
sufficient for its conclusion.
Alimentary my dear canal.
Examples:
Necessary and sufficient:
All Homo sapiens are human is true.
The converse, all humans are Homo
sapiens is also true.
Sufficient but not necessary:
All perch are fish is true.
The converse, all fish are perch is
not true.
Necessary but not sufficient:
All fish are perch is not true.
The converse, all perch are fish is
true.
Neither necessary nor sufficient.
All women are dogs is not true.
The converse, all dogs are women is
not true.
Enough for today!