Lec1.3-4 - L1.3-4 Notes Logic(continued The expression x>3 is not a proposition It contains a free variable However if we specify that x is 1 that

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
L1.3-4 Notes: Logic (continued) The expression x>3 is not a proposition. It contains a free variable. However, if we specify that x is 1, that is, if we bind x to the value 1, then x>3 becomes a proposition. It’s false in this case. Expressions like x>3, containing an unbound or free variable, are called propositional functions . We often denote them like this: P(x): x>3 or sometimes we simply say “Let P(x) denote x>3.” While P(x) is a propositional function, P(9) or P(1) are propositions. P(9) is 9>3 and is True. P(1) is 1>3 and is False. In effect, the substitution of a number for x in the propositional function has the effect of binding the variable x to that number. We can do this with two (or more) variables: Let Q(x,y) be x>y. It is a propositional function in two variables. Q(3,2) is a proposition. It has a truth value. Q(x,2) is a propositional function of one variable. In the above discussion P and Q (by themselves) are called predicates . x and y are variables . Predicates and propositional functions go beyond numbers. We can make them do anything we want. For instance: Let P(x): x murdered Nicole Then P(OJ) is a proposition, namely OJ murdered Nicole. In this case notice that P(3) wouldn’t make sense, any more than OJ>3 would make sense in the earlier cases. Thus we note that predicates and their propositional function require a UNIVERSE OF DISCOURSE (often denoted as U) . U is the collection of entitities which the variables in a propositional function are allowed to be bound to. When you have predicates, propositional functions and variables, there must be a universe of discourse. It may be implicit. It may consist of all possible entitities in the universe. But whatever it is, it must be there, and it is your right to demand to know what the universe of discourse is when shown a predicate used as a propositional function. Let’s move on. Let the Universe U be numbers. Consider the statement: 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
“For every number x, x=3.” Is it a proposition? Can we determine whether it is true or false? The answer is yes, we can. It is a proposition and it is False in this case. We have succeeded in binding the variable by use of the words “For every.” Logically we proceed this way. Let P(x): x>3 We write 2200 xP(x), meaning 2200 x(x>3), meaning in words, for every number x, x>3. More accurately, 2200 x means “For every element in the Universe.” 2200 xP(x) IS TRUE IF AND ONLY IF P(x) IS TRUE FOR EVERY SUBSTITUTION FOR x FROM U. Consider in the same vein the statement: “There is a number x such that x>3.” Is that a proposition? Of course. It’s True. Letting P(x): x>3, we denote this expression as 5 xP(x) or 5 x(x>3) meaning, there is an element in the Universe such that it is larger than 3. 5
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/26/2008 for the course CSC 226 taught by Professor Watkins during the Spring '08 term at N.C. State.

Page1 / 7

Lec1.3-4 - L1.3-4 Notes Logic(continued The expression x>3 is not a proposition It contains a free variable However if we specify that x is 1 that

This preview shows document pages 1 - 3. Sign up to view the full document.

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