Quantifiers and translating English

• Quantifiers tell you the scope of a claim. "For all" (written forall) means a property holds for every element in a specified set, like "for all integers n." "There exists" (written exists) means at least one element has the property, like "there exists an integer n such that..." The set matters: "there exists a real x with x^2 = 2" is true, but "there exists an integer x with x^2 = 2" is false. When translating English into math, identify (1) the set you are talking about, (2) whether the claim is for all or there exists, and (3) the exact property being asserted.
• The order of quantifiers can completely change meaning. Compare "for every x, there exists a y" (forall x exists y) with "there exists a y such that for every x" (exists y forall x). Example: "For every real x, there exists a real y with y > x" is true, but "There exists a real y such that for every real x, y > x" is false (no single number is bigger than every real number). Good proof writing starts with correctly expressing claims, because if you misunderstand the quantifiers, you can prove the wrong thing perfectly.

• Every integer has a successor: forall n in Z, exists m in Z such that m = n + 1.
• Some integer is even: exists n in Z such that n is even.

Translate to symbols.

• 1Every real number has a square that is nonnegative.
• 2Some integer is divisible by 7.
• 3For every integer n, there exists an integer k with n = 2k or n = 2k + 1.
• 4There exists a real x such that for all real y, x <= y.

MCQ

Which translation matches 'Every integer has a successor'?