Question

Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that every positive integer is the sum of the squares of four integers.

Short answer

The statement that every positive integer is the sum of the squares of four integers can be expressed using predicates, quantifiers, logical connectives and mathematical operators.

Step-by-step solution

Expression for every positive integers

Take x is a positive integer x>0. The expression for every positive integer is \(\forall x(x > 0).\)Considering there are four integers a, b, c, d can be expressed as \(\exists a\exists b\exists c\exists d.\)The sum of 4 integers square is x and can be represented as \(x = {a^2} + {b^2} + {c^2} + {d^2}.\)

Expression for the statement

The sum of the squares of four integers can be a positive integer for all positive values of x. It can be represented as,

\(\forall x(x > 0) \to \exists a\exists b\exists c\exists d(x = {a^2} + {b^2} + {c^2} + {d^2}).\)