Question

Is the set \(\theta=\left\\{(X,|X|): X \subseteq \mathbb{Z}_{5}\right\\}\) a function? If so, what is its domain and range?

Short answer

Yes, the set \(\theta\) is a function. The domain of \(\theta\) is the set of all subsets of \(\mathbb{Z}_{5}\), and the range is \(\{0,1,2,3,4,5\}\).

Step-by-step solution

Understand the set

The given set \(\theta\) consists of pairs (X,|X|), where X is a subset of \(\mathbb{Z}_{5}\) and |X| is the cardinality of the set X. Here, \(\mathbb{Z}_{5}\) represents the set \{0, 1, 2, 3, 4\}. Thus, set X can be any possible subset of these 5 elements, and |X| represents the number of elements in set X.

Evaluate the function condition

According to the definition of a function, for \(\theta\) to be a function, each value in the subset of \(\mathbb{Z}_{5}\) should correspond to exactly one value in |X|. Here, for any X, there is exactly one |X|. Therefore, the set \(\theta\) does indeed represent a function.

Define the domain and the range

The domain of a function is the set of all possible input values. In this case, domain of \(\theta\) is the set of all the subsets of \(\mathbb{Z}_{5}\). The range of a function is the set of all possible output values (i.e., the cardinalities of these subsets). Because the subsets of \(\mathbb{Z}_{5}\) can have between 0 to 5 elements, the range is the set of 0 to 5, i.e., \(\{0,1,2,3,4,5\}\).