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Translation Examples
• If a real number is an integer then it is a rational number
• All bytes have eight bits
• No fire trucks are green.
Formal and Informal
• For all polygons p, if p is a square, then p is a rectangle.
• For all squares p, p is a rectangle.
• There exists a number n such that n is prime and n is even.
• There exists a prime number n such that n is even.
Trailing Quantifier
• For all squares p, p is a rectangle.
• p is a rectangle for any square p.
• There exists a prime number n such that n is even.
• n is even for some prime number n.
Implicit Quantification
• If n is a number, then it is a rational number
• If x>2 then x2>4.
• x>2 x2>4
and
• P(x) Q(x) means that the truth set of P(x) is contained in the truth set of Q(x).
• P(x) Q(x) means P(x) and Q(x) have identical truth sets.
Negation of Quantified Statements
• For all x in D, Q(x)
• There exists x in D such that ~Q(x).
• “all are” versus “some are not”
Negation of Quantified Statements
• There exists x in D such that Q(x)
• For all x in D, ~Q(x).
• “some are” versus “all are not”
Try These
• ~(For every prime p, p is odd)
• ~(There exists a triangle T, such that the sum of the angles of T equals 200 degrees)