L08_Quantifiers_print

L08_Quantifiers_print - 1 COMP170 Discrete Mathematical...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 3.2, pp. 104-114 Quantifiers Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 3.2 Variables and Quantifiers • Variables and Universes • Quantifiers • Standard Notation for Quantification • Statements about Variables • Proving Quantified Statements True or False • Negation of Quantified Statements • Implicit Quantification 3 Consider the statement: Variables and Universes (*) m 2 > m Is (*) True or False ? This is an ill-posed question! For some values of m , e.g., m = 2 , (*) is True For other values of m , e.g., m = 1 / 2 , (*) is False In statements such as m 2 > m , variable m is not constrained . Unconstrained variables are called free variables . Each possible value of a free variable gives a new statement. The Truth or Falsehood of this new statement, is determined by substituting in the new value for the variable. 4 • For which values of m is (*) True and for which values is it False ? • For the universe of non-negative integers , the statement is True for every value of m except m = 0 , 1 . • For the universe of real numbers , the statement is True for every value of m except for ≤ m ≤ 1 Two main points: • Clearly state the universe • A statement about a variable can be True for some values of a variable and False for others. Again consider the statement: (*) m 2 > m • This statement is also ill-defined! The answer depends upon which universe we assume 5 3.2 Variables and Quantifiers • Variables and Universes • Quantifiers • Standard Notation for Quantification • Statements about Variables • Proving Quantified Statements True or False • Negation of Quantified Statements • Implicit Quantification 6 Quantifiers The statement • While m 2 > m is True for values such as m =- 3 or m = 9 it is False for m = 0 or m = 1 . • Thus, it is not True that m 2 > m for every integer m , so (*) is False (*) For every integer m , m 2 > m is False . 7 Quantifiers The statement • A phrase like for every integer m that converts a symbolic statement about potentially any member of our universe into a statement about the universe is called a quantifier • A quantifier that asserts a statement about a variable is true for...
View Full Document

Page1 / 27

L08_Quantifiers_print - 1 COMP170 Discrete Mathematical...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online