Negations

• Negation is how you correctly express "the opposite" of a statement, and it is essential for contradiction proofs and for spotting counterexamples. The most important rule is that negation flips quantifiers: the negation of "for all x, P(x)" is "there exists an x such that not P(x)." Symbolically, not (forall x P(x)) is equivalent to exists x not P(x). Similarly, not (exists x P(x)) is equivalent to forall x not P(x). This matches common sense: to deny "everyone passed," you only need "someone failed," and to deny "someone passed," you need "no one passed."
• You also need De Morgan's laws for combining statements. The negation of "P and Q" is "not P or not Q," and the negation of "P or Q" is "not P and not Q." Beginners often negate incorrectly by keeping the same connector. Finally, be careful with inequalities: the negation of "x >= 3" is "x < 3," and the negation of "x > 3" is "x <= 3." Correct negations matter because they tell you exactly what must be shown to disprove a claim or what assumption you take on when proving by contradiction.

• Negate 'For all integers n, n is even': 'There exists an integer n that is not even.'
• Negate 'There exists x such that x^2 = 2' (over integers): 'For all integers x, x^2 != 2.'

Negate each carefully.

• 1For all real x, x^2 >= 1
• 2There exists an integer n such that n is prime and n is even
• 3For every student s, there exists a class c that s has taken
• 4There exists x in R such that for all y in R, x + y = y

MCQ

What is the negation of 'For all real x, x^2 >= 1'?