Math 3000 Quiz #3

Let A be some set of integers.

  1. Rewrite the statement below using logical symbols and the predicates P(x) = "x is an even integer" and Q(x) = "x is a square integer".
  2. State a useful denial of the statement below as an English sentence.
If all the integers of A are either even or squares, then either there is an even integer in A or all the integers of A are squares.
Solution:
  1. (xA)(P(x)Q(x))(xA)(P(x))(xA)(Q(x)).

  2. A denial in symbols is:
    (xA)(P(x)Q(x))(xA)(~P(x))(xA)(~Q(x)).
    Which can be rendered as:
    All the elements of A are either even or squares, but there is no even integer in A and at least one non-square.