R is the set of positive rational numbers

R is a subset of Q, where Q is the set of all rational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers.

- Examples of rational numbers include:

1/2, 3/4, 5/6, 7/8, 9/10, etc.

- Properties of rational numbers:

Rational numbers are closed under addition, subtraction, multiplication, and division operations.

The sum, difference, product, and quotient of two rational numbers is always a rational number.

Rational numbers can be represented on a number line.

E is the set of real numbers

E is the set of all real numbers, which includes both rational and irrational numbers. Real numbers are numbers that can be represented on a number line.

- Examples of real numbers include:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.

0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, etc.

π, √2, √3, √5, etc.

- Properties of real numbers:

Real numbers are closed under addition, subtraction, multiplication, and division operations.

The sum, difference, product, and quotient of two real numbers is always a real number.

Real numbers can be represented on a number line.

Relation between R and E

R is a subset of E, as all rational numbers are real numbers.

- Properties of R and E:

Rational numbers are countable, while real numbers are uncountable.

Rational numbers have a finite or repeating decimal representation, while real numbers can have an infinite and non-repeating decimal representation.

Some real numbers are irrational, which means they cannot be expressed as a ratio of two integers.

Conclusion

In conclusion, R is a subset of Q and E is the set of all real numbers. R is a subset of E, as all rational numbers are real numbers. While rational numbers are countable and have a finite or repeating decimal representation, real numbers are uncountable and can have an infinite and non-repeating decimal representation.