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Chapter 28
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| row | A | NOT (A) |
| 1 | occurs | occurs not |
| 2 | occurs not | occurs |
The following table
| row | A | NOT A | NOT (NOT A) |
| 1 | occurs | occurs not | occurs |
| 2 | occurs not | occurs | occurs not |
| row | situation A | situation B | A and B |
| 1 | occurs | occurs | occurs |
| 2 | occurs | occurs not | occurs not |
| 3 | occurs not | occurs | occurs not |
| 4 | occurs not | occurs not | occurs not |
The situation A and B occurs provided
rows 2, 3 and 4 in the above never occur.
In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not, the situation A and B occurs or occurs not.
* Language Note. The phrase A and B is also labelled (called) the conjunction of the situations A and B. The situation A and B is said to be true when and only when both the situations A and B occur (= are true).
In everyday speech when you use the word or in a phrase like John or Andrew will go to the store, the usual expectation is that only one will go, not both. But there is another use of the word or favored in logic. The word or is employed in the at least one sense (as is done in logic and mathematics). With this sense or usage, the previous phrase is understood in the inclusive sense: John or Andrew, or both, will go to the store. We now proceed and we will use the word or in the at least one sense.
The situation (A or B ) is said to occur if at least one of the two situations A and B occurs. Otherwise, it is said not to occur. This is summarized in the following table.
| row | situation A | situation B | A or B |
| 1 | occurs | occurs | occurs |
| 2 | occurs | occurs not | occurs |
| 3 | occurs not | occurs | occurs |
| 4 | occurs not | occurs not | occurs not |
The situation A or B can be said to occur
provided the situation in row 4 does not occur.
We have to be careful with the word or. Its meaning depends on the speaker and possibly the listener. That is, confusion and ambiguity results when two people in question use the same words but give them different meanings. To eliminate this ambiguity in everyday speech, write and say one of the following:
Any rule which can be stated in the form if a first situation A occurs, then a second situation B occurs, in brief, if A then B or A implies B, is called a one-way implication.
A one-way implication which is never disobeyed is said to hold and to be (always) true. For a one-way implication rule if A then B, we recall the following:
| row | situation A | situation B | if A then B |
| 1 | occurs | occurs | obeyed |
| 2 | occurs | occurs not | disobeyed |
| 3 | occurs not | occurs | not disobeyed |
| 4 | occurs not | occurs not | not disobeyed |
In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not the if-then rule is obeyed, disobeyed, or not disobeyed.
Row 2 represents the situation in which A occurs but B does not. Observe that in this situation, the rule is disobeyed. In the situations represented by the other three rows, the rule is not disobeyed. A one-way implication rule if A then B is said
to be always true,
to always hold
when it is never disobeyed. The one-way implication if A then B is always true when the situation described in row 2 in the above table never occurs.
Remark. If situation A never occurs, the implication rule if A then B is never disobeyed amd it is said to be vacuously true.
A rule which can be stated, or restated, in the form
The first situation A occurs when and only when the second
situation B occurs
or in the form
The first situation A occurs if and only if the second situation B occurs
is called a two-way implication rule. For each two-way implication rule
note that:
The rule is obeyed when both situations occur.
The rule is disobeyed when the first situation A occurs without the second situation B occurring.
The rule is disobeyed when the second situation B occurs without the first situation A.
The rule is not disobeyed when both situations do not occur.
In brief, the two situations in a two-way implication rule must both occur or both must not occur, for the rule to be not disobeyed.
The next table summarizes the above remarks for any two-way implication rule A if and only if B.
| row | situation A | situation B | A if and only if B |
| 1 | occurs | occurs | obeyed |
| 2 | occurs | occurs not | disobeyed |
| 3 | occurs not | occurs | disobeyed |
| 4 | occurs not | occurs not | not disobeyed |
As said before, a two-way implication rule is said to be always true when it is
never disobeyed. This requires that the situations in rows 2 and 3 of the above
table do not occur. That is, the above two-way implication rule A iff B is
true (never disobeyed) provided neither of the situations A and B occurs without
the other.
The converse to the implication rule if A then B is the rule if B then A. Note that interchanging the first and second situation A and B yields the converse to a rule. From this definition or perspective, we see that the converse of the converse is the original rule. Check this.
When we know a rule if A then B is never disobeyed, we have no guarantee that the converse rule if B then A is never disobeyed. The reason for this is as follows. The rule if A then B is true if the situation A never occurs without the situation B. The converse rule if B then A is true if the situation B cannot occur without the situation A.
Reminder. Now we can easily answer the following question: What can we say for sure about the event A when (i) the rule if A then B is never disobeyed, and (ii) the event B occurs? Your answer should be not much, or nothing, without further information.
Next: Chapter 29, Contrapositive Form of Implication and Conditionals IF A THEN B
Three Skills for Algebra
www.whyslopes.com
Foreword, Chapters
& Appendices
| Real Player Videos Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14. |
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
with stick diagrams.
(iv) 5a + 16 = 3a+ 24
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
without
Stick Diagrams
essentially one unknown
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