Question

Let r1 and r2 be relations on a set a represented by the matrices mr1 = ⎡ ⎣ 0 1 0 1 1 1 1 0 0 ⎤ ⎦ and mr2 = ⎡ ⎣ 0 1 0 0 1 1 1 1 1 ⎤ ⎦. find the matrices that represent
a.r1 ∪ r2.
b.r1 ∩ r2.
c.r2 ◦r1.
d.r1 ◦r1.
e.r1 ⊕ r2.

Answer 1

Given:
$r1 = \left[\begin{array}{ccc}0&1&0\\1&1&1\\1&0&0\end{array}\right] \,\, and \,\, r2= \left[\begin{array}{ccc}0&1&0\\0&1&1\\1&1&1\end{array}\right]$

Part a. r1 ∪ r2
This assembles matrix elements from either r1 and/or r2.
$r1 \cup r2 = \left[\begin{array}{ccc}0&1&0\\1&1&1\\1&1&1\end{array}\right]$

Part b. r1 ∩ r2
This assembles matrix elements common to both r1 and r2.
$r1 \cap r2 = \left[\begin{array}{ccc}0&1&0\\0&1&1\\1&0&0\end{array}\right]$

Part c. r2 . r1
$r2 \circ r1 = \left[\begin{array}{ccc}0&1&0\\0&1&1\\1&1&1\end{array}\right] \left[\begin{array}{ccc}0&1&0\\1&1&1\\1&0&0\end{array}\right] = \left[\begin{array}{ccc}1&1&1\\2&1&1\\2&2&1\end{array}\right]$

Part d. r1 . r1
$r1 \circ r1 = \left[\begin{array}{ccc}0&1&0\\1&1&1\\1&0&0\end{array}\right] \left[\begin{array}{ccc}0&1&0\\1&1&1\\1&0&0\end{array}\right] = \left[\begin{array}{ccc}1&1&1\\2&2&1\\0&1&0\end{array}\right]$

Part e. r1 ⊕ r2 (Direct sum)
$r1 \oplus r2 = \begin{bmatrix} r1&0\\0&r2\end{bmatrix} = \begin{bmatrix} 0&1&0 &0&0&0\\ 1&1&1 &0&0&0\\ 1&0&0 &0&0&0\\ 0&0&0 &0&1&0\\0&0&0 &0&1&1\\ 0&0&0 &1&1&1\end{bmatrix}$