Consider the matrices
a =
3 1 −
1
2
2
2
3
4
and b =
1 0
0 1
0 0

a. multiply a and b or explain why you cannot.
b. would you consider b to be an identity matrix for a? why or why not?
c. would you consider = 0
0 1
or =
1 0 0
0 1 0
0 0 1
an identity matrix for a? why or why not?

Vacuous proof is used.

Step-by-step explanation:

Given:

Proposition p(n) :

"if n is a positive integer greater than 1, then n² > n"

To prove:

Prove the proposition p (0)

Solution:

Using the proposition p(n) the proposition p(0) becomes:

p(0) = "if 0 is a positive integer greater than 1, then 0² > 0"

The proposition that "0 is a positive integer greater than 1" is false

Since the premises "if 0 is a positive integer greater than 1" is false this means the overall proposition/ statement is true.

Thus this is the vacuous proof which states that:

if a premise p ("0 is a positive integer greater than 1") is false then the implication or conditional statement p->q ("if n is a positive integer greater than 1, then n² > n") is trivially true.

So in vacuous proof, the implication i.e."if n is a positive integer greater than 1, then n2 > n." is only true when the antecedent i.e. "0 is a positive integer greater than 1" cannot be satisfied.