Consider the following relations. Which is a one-to-one function?

• A. $\left\{(1,4),(2,5),(3,6),(4,7)\right\}$
• B. $\left\{(1,2),(1,3),(2,4)\right\}$
• C. $\left\{\right(2,3),(3,3),(4,5\left)\right\}$
• D. $\left\{\right(1,1),(2,2),(3,1),(9,9\left)\right\}$

Answer: (A)

Verified step by step guidance

1. Recall that a function is one-to-one (injective) if each element in the domain maps to a unique element in the range, meaning no two different domain values share the same range value.
2. Examine the first relation: \(\left\lbrace (1,4), (2,5), (3,6), (4,7) \right\rbrace\). Check if any range values repeat. Since 4, 5, 6, and 7 are all distinct, this relation is one-to-one.
3. Look at the second relation: \(\left\lbrace (1,2), (1,3), (2,4) \right\rbrace\). Notice that the domain value 1 maps to both 2 and 3, which violates the definition of a function (each input must have exactly one output), so it is not a function, and thus not one-to-one.
4. Check the third relation: \(\left\lbrace (2,3), (3,3), (4,5) \right\rbrace\). Here, the range value 3 is repeated for different domain values 2 and 3, so it is a function but not one-to-one.
5. Finally, analyze the fourth relation: \(\left\lbrace (1,1), (2,2), (3,1), (9,9) \right\rbrace\). The range value 1 appears twice for domain values 1 and 3, so this function is not one-to-one.