COMP SCI 241 Discrete Mathematics I
Homework # 13
Solutions
Spring 2010
F. Baulieu
Let our domain of discourse be the set of all 5  5 real matrices.
Let
P(X) : X is nonsingular
Q(X) : X = X-1
R(X) : X is diagonal
1) Translate into quantified symbolic form:
"Every 5  5 real matrix is singular or is its own inverse.".
̅̅̅̅̅̅̅ ∨ 𝑄(𝑋))
(∀𝑋)(𝑃(𝑋)
2) Prove or disprove the statement in (1).
We disprove it by counterexample.
2
0
0
0
[0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1/2
0
0
0 has inverse 0
0
0
[ 0
1]
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0 so is neither singular nor its own inverse.
0
1]
3) Translate into easily understood mathematical prose:
X R (X)  P(X) 
Every diagonal 5x5 real matrix is nonsingular
4) Translate into easily understood mathematical prose:
X R (X)  P(X)  Q(X) 
There is a diagonal nonsingular 5x5 real matrix that is not its own inverse
5) Prove or disprove the statement in (4)
We prove it by displaying such a beast: See problem 2.