Which of the following candidates are binary or unary operations on the given sets? For

Chapter 4, Problem 40

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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.

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.

ANSWER:

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 \(\rightarrow\) 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.

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