Solution Found!
Which of the following candidates are binary or unary operations on the given sets? For
Chapter 4, Problem 40(choose chapter or problem)
Which of the following candidates are binary or unary operations on the given sets? For those that are not, where do they fail?
a. \(x \#=x^{2} ; S=\mathbb{Z}\)
b. \(\begin{array}{l|lll}
\circ & 1 & 2 & 3 \\
\hline 1 & 1 & 2 & 3 \\
2 & 2 & 3 & 4
\end{array} \quad S=\{1,2,3\}\)
c. \(x \circ y\) = that fraction, x or y, with the smaller denominator; S = set of all fractions.
d. \(x \circ y\) = that person, x or y, whose name appears first in an alphabetical sort; S = set of 10 people with different names.
Questions & Answers
QUESTION:
Which of the following candidates are binary or unary operations on the given sets? For those that are not, where do they fail?
a. \(x \#=x^{2} ; S=\mathbb{Z}\)
b. \(\begin{array}{l|lll}
\circ & 1 & 2 & 3 \\
\hline 1 & 1 & 2 & 3 \\
2 & 2 & 3 & 4
\end{array} \quad S=\{1,2,3\}\)
c. \(x \circ y\) = that fraction, x or y, with the smaller denominator; S = set of all fractions.
d. \(x \circ y\) = that person, x or y, whose name appears first in an alphabetical sort; S = set of 10 people with different names.
Step 1 of 4
The given set \(A\) is a unary operator if and only if #: A \(\rightarrow\) A is well defined.
The given set \(A\) is a binary operator if and only if #: A x A \(\rightarrow\) A is well defined.
(a) Here, X # = \(x^2\), \(S=Z\)
The operator is defined from #: Z \(\rightarrow\) Z
And x # = \(x^2\) is well defined because if \( x \epsilon z\) then \(x^2 \epsilon Z\)
So, # is a unary operator.