Let's go through each statement to see if it is true or false.
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Integers are always whole numbers
This is true. Integers are numbers that don't have any fractional or decimal portion
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All whole numbers are rational
This is also true. Let's say we had the whole number 7. We can write it as 7/1 which is rational. A rational number is any number that can be written as a fraction of two integers. So this means any whole number can be converted to a rational number format.
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Integers include all rational members
This is false. It is true that all integers are rational, but there is more to the story than just integers when it comes to listing all rational numbers.
For instance, the number 1/2 is rational, but it is not an integer.
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Rational numbers do not include integers.
This is false. As stated earlier, any integer can be placed over 1 to form a rational number. So the rational numbers do include integers, among non-whole number fractions as well.
