Universal and Existential Quantifiers in Equations

For all $x$, there exists a $y$ such that $x - 3y = 0$. Is this statement true or false?

This problem provides an excellent chance to explore the use of quantifiers in mathematical logic, a fundamental tool used throughout mathematics and computer science. Here, we are dealing with the universal quantifier 'for all' and the existential quantifier 'there exists.' When these quantifiers are combined within a statement involving equations, it's crucial to understand what is being claimed. In this particular problem, we are asked to determine whether a given statement about lines in plane algebra is universally true or not, which requires us to carefully consider the meaning of each quantifier as it applies to the variables involved.

The statement given can be analyzed by understanding what each term demands. For every value of x, the statement claims that there exists a corresponding y that satisfies the equation $x = 3y$. By manipulating this equation, it's understood that for each x, the value y must be such that x is exactly three times y. This condition can only be true for x values that are multiples of three. This logical exploration involves examining the nature of the integers, divisibility, and how manipulating equations helps us see whether a universal quantifier can indeed be satisfied for all cases. Through this process, students get a chance to reinforce their understanding of proof techniques, logical reasoning, and the structuring of mathematical arguments.