Conditional Statements
Theorems are usually given in the form
of conditional statements. Look at
conditional statements in general. They
all have the form: If P, then Q where P is
called the hypothesis and Q is the
conclusion.
Example: If it snows tomorrow, there will
be no class.
P = it snows tomorrow.
Q = there is no class.
The contrapositive of the statement is:
If not Q, then not P.
If there is class tomorrow, then it will not
be snowing.
The contrapositive of a true statement is
true.
The converse of the statement is:
If Q then P:
If there is no class tomorrow, then it will
snow.
NOT A TRUE STATEMENT.
In general, the converse of a true
statement is not true.
The inverse of the statement is:
If not P, then not Q.
If it does not snow tomorrow, then there
will be class.
False.
In general, the inverse of a true
statement is not true.
Example: If a rectangle is equilateral,
then it is a square.
Contrapositive: If a rectangle is not a
square, then it is not equilateral. (true)
Converse: If a rectangle is a square,
then it is equilateral. (true)
Inverse: If a rectangle is not equilateral,
then it is not a square. (True)
Theorem 1.7: An indexed set
S  v1 , v 2 ,..., v p  of two or more vectors is
linearly dependent iff at least one of the
vectors in S is a linear combo of the
others.
In fact, if S is linearly dependent and v1 ≠
0, then some vector vj
(j ≥ 2) is a linear combo of the preceding
vectors.
The statement: If an indexed set
S  v1 , v 2 ,..., v p  of two or more vectors is
linearly dependent, then at least one of
the vectors in S is a linear combo of the
others. (True)
The contrapositive: If none of the vectors
in S is a linear combo of the others, the
indexed set S  v1 , v 2 ,..., v p  of two or more
vectors is linearly independent. (True)
Since Theorem 1.7 says “if and only if”,
the converse and the inverse are also
true.
Converse: If at least one of the vectors in
S is a linear combination of the others,
the indexed set S  v1 , v 2 ,..., v p  of two or
more vectors is linearly dependent.
(True)
Inverse: If an indexed set S  v1 , v 2 ,..., v p 
of two or more vectors is linearly
independent, then none of the vectors in
S is a linear combination of the others.
Theorem 1.8: If a set contains more
vectors than there are entries in each
vector, then the set is linearly dependent
(i.e. any set v1 , v 2 ,, v p  is linearly
dependent if p > n).
Only the contrapositive is true: If a set is
linearly independent, then the set
contains no more vectors than there are
entries in each vector.
Theorem 9: If a set of vectors
S  v1 , v 2 ,, v p  contains the zero vector,
it is linearly dependent.
Only the contrapositive is true: If a set of
vectors S  v1 , v 2 ,, v p  is linearly
independent, the set does not contain
the zero vector.
Negation of statements
In general, to negate a statement you
insert the word not.
Ex: We are in building 28.
Negation: We are not in building 28.
Ex: The sky is not green.
Neg: The sky is green.
It gets more complicated when you are
making a general statement about a
group of things.
Ex: Men are taller than women.
Means: all men are taller than all women.
All it takes to negate this is one
exception:
There is at least one man who is shorter
than at least one woman.
►To negate a statement using “all”, use
“there exists one” that does not meet the
requirements.
Ex: No dog can sing.
Or All dogs do not sing.
Neg: there is at least one dog that can
sing.
Ex: Some birds can swim.
Or there is at least one bird that can
swim.
Neg: There is not at least one bird that
can swim. Or No birds can swim.
Note: a statement and its negation
cannot both be true.
Converse, Inverse and Contrapositive
If it snows tomorrow, there will be no
class.
P = it snows tomorrow.
Q = there is no class.
The converse of the statement is:
If Q then P: If there is no class tomorrow,
then it will snow.
The inverse of the statement is:
If P, then not Q.
If it snows tomorrow, then there will be
class.
The contrapositive of the statement is:
If not Q, then not P.
If there is class tomorrow, then it will not
be snowing.