Answer:
Yes, the set of rational expressions is closed under multiplication.
Explanation:
A rational expression is a fraction of polynomials, where the numerator and denominator are both polynomials with coefficients in some field, such as rational numbers. When we multiply two rational expressions, we reproduce two fractions of polynomials.
To multiply two fractions, we multiply their numerators and denominators separately and then simplify the result if possible. When we multiply the numerators and denominators of two rational expressions, we get new polynomials, which are still polynomials with coefficients in the same field. Therefore, the multiplication result is still reasonable, and the set of rational expressions is closed under multiplication.
For example, consider the following two rational expressions:
(a + 1) / (b + 2) and (c - 3) / (d - 4)
Their product is:
[(a + 1) / (b + 2)] * [(c - 3) / (d - 4)]
= (a + 1)(c - 3) / (b + 2)(d - 4)
The result is still rational since the numerator and denominator are both polynomials with coefficients in the same field. Therefore, the set of logical expressions is closed under multiplication.