Final answer:
The statement is translated into symbolic logic with quantifiers, and a direct proof is provided showing that the sum of two rational numbers is rational by expressing the sum as the ratio of integers.
Step-by-step explanation:
Statement Formulation Using Quantifiers and Symbolic Logic
The statement "If x and y are rational numbers, then x + y is rational" can be written using quantifiers and symbolic logic as: ∀x, ∀y (R(x) ∧ R(y) → R(x + y)). This indicates that for all x and for all y, if x is a rational number and y is a rational number, then the sum x + y is also a rational number.
Proof Using Direct Proof
To prove this statement using direct proof, assume that x and y are rational numbers. By definition, this means there exist integers a, b, c, and d such that b and d are not zero and x = a/b, y = c/d. When adding these, x + y = (a/b) + (c/d) = (ad + bc)/(bd). Since integers are closed under multiplication and addition, ad + bc and bd are integers, with bd not equal to zero. Therefore, x + y can be expressed as the ratio of two integers with a non-zero denominator, which by definition makes it a rational number. Hence, the original statement is proven to be true.