Min-Max
The \(Min(a,b)\) and \(Max(c,d)\) in the reals are both commutative and associative. (The proof is left to the reader). Check also the Indicator Function \(\mathbb{1}_A(x)\).
A binary operation \( \circ \) on a set \( A \) is a function \[\circ :A \times A \to A\], it means that the function takes an ordered pair and creates another element of \(A\). This property is known as closure. The multiplication and addition in the set of integers are closed, it means that, another integer is produced by those operations. However, the division \(x:y\)is not closed in the integers for the value is undefined for \( y = 0 \) and not all divisions such as \( \dfrac{22}{7} \) or \(1:2\) are integers, they belong to rational numbers. By the way, the first fraction is a rough approximation of the number \[ \pi \approx \dfrac{22}{7} = 3.1428... \],
How about commutativity and associativity. An operation \( \circ \) is associative if \[(a \circ b) \circ c = a \circ (b \circ c)\]
and it is commutative if \[a \circ b = b \circ a\]. For example, let \[S = \{0,1\}\] and \[ \oplus: S \times S \to S\] is
the exclusive-or (xor) operation in a Boolean algebra. This operation is clearly commutative as there are
two cases:
case 1: \(a = b\) \[0 = a \oplus b = b \oplus a = 0\] and
case 2: \(a \not= b\), giving
\[1 = a \oplus b = b \oplus a = 1\]. As for the associative, we have \[(a \oplus b) \oplus c = \{0,1\} \oplus c = a \oplus \{0,1\}\]
\[ 0 \oplus c \vee 1 \oplus \ c \ = a \oplus 0 \vee \ a \ \oplus 1\], as they both are contradictories both sides
evaluate to 1. The another way of seeing the association and commutation is by means of a table.
$$
\begin{array}{c|cc}
\oplus & 0 & 1 \\ \hline
0 & 0 & 1 \\
1 & 1 & 0
\end{array}
$$
$$
\begin{array}{ccc|cc|cc}
a & b & c & (a \oplus b) & \mathbf{(a \oplus b) \oplus c} & (b \oplus c) & \mathbf{a \oplus (b \oplus c)} \\ \hline
0 & 0 & 0 & 0 & \mathbf{0} & 0 & \mathbf{0} \\
0 & 0 & 1 & 0 & \mathbf{1} & 1 & \mathbf{1} \\
0 & 1 & 0 & 1 & \mathbf{1} & 1 & \mathbf{1} \\
0 & 1 & 1 & 1 & \mathbf{0} & 0 & \mathbf{0} \\
1 & 0 & 0 & 1 & \mathbf{1} & 0 & \mathbf{1} \\
1 & 0 & 1 & 1 & \mathbf{0} & 1 & \mathbf{0} \\
1 & 1 & 0 & 0 & \mathbf{0} & 1 & \mathbf{0} \\
1 & 1 & 1 & 0 & \mathbf{1} & 0 & \mathbf{1}
\end{array}
$$
Min-MaxThe \(Min(a,b)\) and \(Max(c,d)\) in the reals are both commutative and associative. (The proof is left to the reader). Check also the Indicator Function \(\mathbb{1}_A(x)\). |
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