Question

y = {(2, 6), (3, 1, 4), (1), (5, 6)} is y a function and why?
no, the elements of the set of y are not all ordered pairs.
yes, there is more than one ordered pair in this list.
no, there is a limited number of ordered pairs in this list.
yes, no two ordered pairs in this list has the same second element.

Explanation:

To determine if \( y \) is a function, we first recall the definition of a function as a relation where each element of the domain (first element of ordered pairs) is associated with exactly one element of the codomain (second element of ordered pairs), and importantly, a function must be a set of ordered pairs.

Looking at the set \( y = \{(2, 6), (3, 1, 4), (1), (5, 6)\} \), we check the elements:

• \( (2, 6) \) is an ordered pair.
• \( (3, 1, 4) \) is not an ordered pair (it has three elements, while an ordered pair has two).
• \( (1) \) is not an ordered pair (it has one element).
• \( (5, 6) \) is an ordered pair.

For a relation to be a function, all its elements must be ordered pairs (since a function is a type of relation, and relations are sets of ordered pairs). Here, not all elements of \( y \) are ordered pairs (due to \( (3, 1, 4) \) and \( (1) \)), so \( y \) is not a function.

Now let's analyze the options:

• Option 1: "No, the elements of the set of \( y \) are not all ordered pairs." This matches our reasoning because \( (3, 1, 4) \) and \( (1) \) are not ordered pairs.
• Option 2: "Yes, there is more than one ordered pair in this list." The presence of multiple ordered pairs is not sufficient for a function; the key is that all elements must be ordered pairs and each domain element maps to one codomain element. So this is incorrect.
• Option 3: "No, there is a limited number of ordered pairs in this list." The number of ordered pairs (limited or not) is not the reason for \( y \) not being a function. The issue is the presence of non - ordered pairs. So this is incorrect.
• Option 4: "Yes, no two ordered pairs in this list has the same second element." First, \( y \) is not a function because of non - ordered pairs, and also, the condition of having unique second elements is for one - to - one functions (a stricter condition than just being a function), and it's not relevant here as the main issue is the non - ordered pairs. So this is incorrect.

Answer:

A. No, the elements of the set of \( y \) are not all ordered pairs.