Question

four relations are shown below. click on the tabs to see them.
relation 1 relation 2 relation 3 relation 4
{(b, n), (n, b), (g, g), (d, b)}
check the box corresponding to each relation that represents a function.
☑️ relation 1 ☐ relation 2 ☐ relation 3 ☐ relation 4 ☐ none of the relations

Explanation:

To determine if a relation is a function, we use the definition: a relation is a function if every input (first element of each ordered pair) has exactly one output (second element of each ordered pair).

Analyzing Relation 3:

The ordered pairs in Relation 3 are \(\{(b, n), (n, b), (g, g), (d, b)\}\).

• Input \(b\) has output \(n\).
• Input \(n\) has output \(b\).
• Input \(g\) has output \(g\).
• Input \(d\) has output \(b\).

Each input (\(b\), \(n\), \(g\), \(d\)) appears exactly once in the first position of the ordered pairs, so each input has exactly one output. Thus, Relation 3 is a function.

Note on the Given Check (Relation 1):

Since the problem involves checking which relations are functions, and we analyzed Relation 3 (the visible one), we also need to consider that if Relation 1 was initially checked, we assume (from the problem’s context) that Relation 1 is also a function (though its pairs are not shown here). However, based on the visible Relation 3, it is a function.

Answer:

• Relation 1 (if it satisfies the function definition, as per the initial check)
• Relation 3 (as analyzed, it is a function)

(Assuming Relation 1 is valid, the correct checked boxes would be:
\(\boldsymbol{\checkmark}\) Relation 1
\(\boldsymbol{\square}\) Relation 2
\(\boldsymbol{\checkmark}\) Relation 3
\(\boldsymbol{\square}\) Relation 4
\(\boldsymbol{\square}\) None of the relations

But since only Relation 3’s pairs are shown, and it is a function, at minimum, Relation 3 is a function. If Relation 1 is also a function (per the initial check), it is included too.)