1549901369-Elements_of_Real_Analysis__Denlinger_

A. l The Logic of Propositions 585

Examples A.1.3 Some conjunctions:

(a) I ate a snack and then I went to bed.

(b) "I like cheese and crackers" is a conjunction when it is intended to mean

"I like cheese and I like crackers," but it is not a conjunction when it is

intended to mean "I like cheese with crackers."

(c) "l < 3 < 7" is the conjunction "1 < 3 and 3 < 7." D

Definition A.1.4 The disjunction "P or Q" is symbolized P V Q and is

defined by the truth-table:

Table A.2

p Q P V Q

T T T

T F T

F T T

F F F

Notice that, according to Table A .2, P V Q is true when both P and Q
are true. This tells us that we are using the inclusive "or" here. That is , P V Q
includes the possibility that P and Q are both true. It is false only when both
P and Q are false. Sometimes in everyday usage, "or" is used in a different
sense, the exclusive sense, to exclude that possibility. For example, a person
saying, "Either you believe me or you don't" would most likely be intending
the exclusive meaning of "or." In ordinary conversation it is up to the user of
the word "or" to decide what he or she means. But in logic and mathematics,
we cannot allow this ambiguity. Thus, we agree to use only the inclusive "or. "
For us "or" will always be equivalent to "and/ or."

Examples A.1.5 Some disjunctions:

(a) I will see you tonight or I will phone you. (The inclusive "or" allows
that I could do both.)

(b) 7 2: 3. (This is the disjunction "7 > 3 or 7 = 3.")
( c) x = 1 or x = 2. ( 0 bserve that the solution of the equation "x^2 - 3x + 2 =
0" is the disjunction "x = 1 or x = 2 ," not the conjunction "x = 1 and x = 2."
Many students use "and" here, when the proper connective is "or." In fact, the
conjunction here would be incorrect , even false.) D