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Flashcards: Symbolic Logic

Looking at the following truth table, find the missing operator if

$\text{result}=p\bigstar q$ .

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&F\\ T&T&T\\ F&T&F\\ F&F&T\\ \hline \end{tabular}$

$\vee$

$\wedge$

$\rightarrow$

$\lnot$

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Flashcards: Symbolic Logic

Looking at the following truth table, find the missing operator if

$\text{result}=p\bigstar q$ .

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&F\\ T&T&T\\ F&T&F\\ F&F&T\\ \hline \end{tabular}$

Your answer:

$\vee$

Your answer:

$\wedge$

Your answer:

$\lnot$

Your answer:

Correct answer:

$\rightarrow$

Explanation:

To help solve for the missing operator in this truth table, first recall the different operators and there meanings.

$\\\vee:\text{Or} \\\wedge:\text{And} \\\lnot:\text{Not} \\\rightarrow:\text{Implies} \\\Leftrightarrow:\text{Equivalence}$

In truth tables when the "or" operator is used translates to, either and (the constants) being true. When the "and" operator is used that means that for the result to hold true both the constants must be true. The "not" operator negates the answer. The "implies" that the first constant results in the second constant . Lastly, the "equivalency" operator signifies that both constants are the same.

Looking at the truth table,

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&F\\ T&T&T\\ F&T&F\\ F&F&T\\ \hline \end{tabular}$

and result in a true statement whenever the first constant is the same as the second constant. Therefore, the missing operator is "implies".

In mathematical terms the missing operator is $\rightarrow$ .

Flashcards: Symbolic Logic

Identify the complex sentence of the following statement:

Sally has a basketball and she sells it to her friend Bob.

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Sold(Sally, Basketball)}$

$\lnot\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \text{Sold(Sally, Basketball)}$

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

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Flashcards: Symbolic Logic

Sally has a basketball and she sells it to her friend Bob.

Your answer:

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Sold(Sally, Basketball)}$

Your answer:

$\lnot\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

Your answer:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

Your answer:

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \text{Sold(Sally, Basketball)}$

Correct answer:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

Explanation:

First-order logic statements can be described in complex sentences by using logic symbols.

Recall the following logic symbols.

$\lnot$ means "not"

$\Rightarrow$ means "implies"

$\vee$ means "or"

$\wedge$ means "and"

$\Leftrightarrow$ means "equivalent"

For this particular problem the starting sentence is,

"Sally has a basketball and she sells it to her friend Bob."

First, identify the first-order statements and write them in symbolic form. This particular sentence has two first-order statements.

Statement 1: Sally has a basketball

$\text{Owns(Sally, basketball)}$

Statement 2: Sally sells her basketball to her friend Bob.

$\text{Sold(Sally, basketball, Bob)}$

To combine these statements into one complex sentence, it needs to be understood that once Sally sells her basketball she no longer has it therefore, the statement becomes:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

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