Logical Connectives
- Thread starter sar22bier
- Start date
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JeffM
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sar22bier said:
There is no need to post the same question three times.
There is a very good article on truth tables in wikipedia.
The letters in this branch of mathematics stand for sentences (propositions).
Let's start with your second problem. You are given:
p T
q T
r F
If p is a true statement, which is what you have been told to assume, then ~p, meaning the opposite statement, is false. You can always get your mind around this stuff by substituting a specific example. Suppose p is "Tom is a cat." You are told that p is true. So, in that case, the opposite statement that "Tom is not a cat" must be false. Right? OK So the truth table can be expanded.
p T
q T
r F
~p F
Now what about r. You are told that r is false. Let's say r is "Pete is a dog." But it is a false statement. So what is ~r. Why it is the statement that "Pete is not a dog." And that is true. So the truth table can be expanded.
p T
q T
r F
~p F
~r T
Now (a V b) means that AT LEAST ONE of statements a or b is true. A compound statement may be true or false. SUPPOSE for example that you are a girl in 11th grade. Then the compound statement "I am a girl or I am in 10th grade" is true because one is true and one is at least one. The compound statement "I am a boy or I am in 11th grade" is also true for the same reason. The compound statement "I am a girl or I am in 11th grade" is also true because then both are true and 2 is at least 1. But the compound statement "I am a boy or I am in 12th grade" is false. But what about (q V ~r). Well according to the truth table both q and ~r are true so certainly at least one of them is true. So the truth table can be expanded again.
p T
q T
r F
~p F
~r T
(q V ~r) T
Now we come to the tricky part of this problem. For propositions of the form a --> b, we say that it is true UNLESS a is true and b is false. Suppose I say "If Roger is a horse, then Roger is in the barn." If it is true that Roger is a horse and also true that he is in the barn, then the statement is obviously true. If it is true that Roger is a horse but false that Roger is in the barn, then the statement is obviously false. But if it is false that Roger is a horse, then the statement does not say anything about where Roger is. We could say that such a statement is meaningless, but it certainly is not false. So we say it is true rather than meaningless.
So the truth table can be extended one last time to cover [~p --> (q V ~r)]. Remember that ~p is false.
p T
q T
r F
~p F
~r T
(q V ~r) T
[~p --> (q v ~r)] T
Hello, sar22bier!
. . \(\displaystyle \begin{array}{|c|c|c|| c|c|c|c|c|} p & q & r & \sim\!p & \to & (q & \vee & \sim\!r) \\ \hline \hline T & T & F & F & {\bf T} & T & T & T \\ \hline\hline & & & _1 & _3 & _1 & _2 & _1 \\ \hline \end{array}\)
. . \(\displaystyle \begin{array}{|c|c||c|c|c|c|c|} p & q & (\sim\!p & \vee & q) & \to & p \\ \hline\hline T & F & F & T & T & {\bf T} & T \\ T & F & F & F & F & {\bf T} & T \\ F & T & T & T & T & {\bf F} & F \\ F & F & T & T & F & {\bf T} & F \\ \hline\hline & & _1 & _2 & _1 & _3 & _1 \\ \hline \end{array}\)\(\displaystyle \text{1. Construct the truth table for: }\;(\sim\!p\,\vee\,q)\:\to\\)
\(\displaystyle \text{2. Given }p\text{ is true, }q\text{ is true, and }r\text{ is false,}\)
. . . \(\displaystyle \text{find the truth value of the statement: }\:\sim\!p \:\to\
. . \(\displaystyle \begin{array}{|c|c|c|| c|c|c|c|c|} p & q & r & \sim\!p & \to & (q & \vee & \sim\!r) \\ \hline \hline T & T & F & F & {\bf T} & T & T & T \\ \hline\hline & & & _1 & _3 & _1 & _2 & _1 \\ \hline \end{array}\)