Answers: statements 1, 2, 6 and 7
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Step-by-step explanation:
I'll briefly go through each statement to mention why it's true or false.
- This one is a bit tricky. Some math books define the natural numbers to be N = {1, 2, 3, ...} where 0 isn't included, while other textbooks include 0. The inclusion/exclusion of zero seems arbitrary and depends on other context of the math theorems used. The set of whole numbers is more consistent and it is W = {0, 1, 2, 3, ...} basically it's the definition of natural numbers where 0 is included. So if 0 is defined to be a natural number in your textbook, then statement 1 is true. Otherwise, it's false.
- This is true. A rational number like 2/3 isn't a whole number, but something like 3/1 = 3 is a whole number. Any whole number can be written as itself over 1.
- This is false. An integer like -7 is not in the set of natural numbers.
- This is false. The very definition of "irrational" means "not rational". The two types of numbers are complete opposites. An irrational number is one where we cannot write it as a ratio of integers. A famous example is pi = 3.14... as we can't write it as a fraction of integers. An approximation could be something like 22/7, but we don't capture the full picture of what pi really is.
- This is false. Refer to question 4 above. The set of rational numbers vs irrational numbers do not overlap at all.
- This is true. Refer to questions 4 and 5.
- This is true. We can write any integer as itself over 1. For example, the integer -12 is the same as -12/1 to show that it is rational. This idea is similar to what I mentioned in question 2.
After going through all seven statements, we definitely know that statements 2, 6, 7 are true. Statements 3,4,5 are definitely false.
Statement 1 is the only thing left. If your teacher mentioned 4 true statements, then statement 1 has to be true. This in turn means that your teacher is defining 0 to be a natural number.