Final answer:
The set R, which is defined by numbers that square to a units digit of 1 or 9, includes numbers that are not necessarily prime, but must be odd. Negative numbers can also be included in R since their square also produces a positive number.
Step-by-step explanation:
The question asks about the set R, which consists of all numbers that when squared have a units digit of 1 or 9. To investigate which statements must be true about set R, consider the nature of numbers when they are squared and what their units digit can be.
Firstly, let's address statement i, that all members of R are prime. This is not necessarily true; consider the number 4, which is not prime, but when squared (4² = 16), the resulting units digit is 6, not 1 or 9. Therefore, not all numbers that square to a units digit of 1 or 9 are prime.
Next, statement ii claims all members of R are odd. If we square an odd number (like 1, 3, 5, 7, or 9), the units digit can indeed be 1 or 9 in some cases (1²=1, 3²=9, ...). Odd numbers do not produce an even units digit when squared, so all members of R must indeed be odd.
Lastly, statement iii says there are no negative numbers in R. This statement is incorrect, as squaring a negative number results in a positive number. Thus, negative numbers can have units digits of 1 or 9 when squared (for example, (-3)² = 9). Therefore, R can include negative numbers.
The only statement that must be absolutely true about set R is that all members of R are odd. Therefore, the correct answer is ii only.