Learning Objectives
- Commas in Logical Statements
- Order of Logical Operations
- Create truth tables for complex statements using all operations (negations, conjunctions, disjunctions, conditionals, biconditionals)
When combining operations in logic, commas are important.
Commas in Logical Statements
When we translate English statements into logical symbols, commas are used to show where the parentheses are placed.
An easy way to visualize it is to think in terms of the symbols.
Example 1: Suppose P, Q, and R are simple statements. Now consider the compound statement:
P and Q, or R.
The comma comes after the Q, so “P and Q” is inside parentheses.
(P and Q), or R.
Or, if we write using all symbols, P and Q, or R becomes:
[latex]\left(P\land Q\right)\lor{R}[/latex].
Example 2: Suppose P, Q, and R are simple statements. Now consider the compound statement:
If P then Q, if and only if R.
The comma comes after the Q, so “if P then Q” is inside parentheses. Using symbols that looks like:
[latex]\left(P\rightarrow Q\right)\leftrightarrow R[/latex].
Example 3: Suppose P, Q, and R are simple statements. Now consider the compound statement:
If P, then Q if and only if R.
Here the comma comes after the P, so “Q if and only if R” is inside parentheses. Using symbols that looks like:
[latex]P\rightarrow \left(Q\leftrightarrow R\right)[/latex].
Now let’s look at examples using English statements.
Example 4: Translate the English statement into logical symbols.
Let P: The air is cold Q: The humidity is high R: I will wear shorts
The air is cold and the humidity is high, or I will wear shorts.
Solution: [latex]\left(P\land Q\right)\lor{R}[/latex]
Example 5: Translate the English statement into logical symbols.
Let P: The bookstore is closed Q: The parking lot is full R: the cafeteria is serving pizza
The bookstore is not closed and the parking lot is not full, or the cafeteria is serving pizza.
Solution: [latex]\left({\sim P}\land \sim Q\right)\lor{R}[/latex]
Example 6: Translate the English statement into logical symbols.
Let P: The bookstore is closed Q: The parking lot is full R: the cafeteria is serving pizza
The bookstore is not closed, and the parking lot is not full or the cafeteria is serving pizza.
Solution: [latex]{\sim P}\land \left( \sim Q\lor {R}\right)[/latex]
Example 7: Translate the English statement into logical symbols.
Let P: The bookstore is closed Q: The parking lot is full R: the cafeteria is serving pizza
If the bookstore is closed, then the parking lot is full if and only if the cafeteria is serving pizza.
Solution: [latex]P\rightarrow \left(Q \leftrightarrow R\right)[/latex]
Example 8: Translate the English statement into logical symbols.
Let P: The bookstore is closed Q: The parking lot is full R: the cafeteria is serving pizza
If the bookstore is closed then the parking lot is full, if and only if the cafeteria is serving pizza.
Solution: [latex]\left(P\rightarrow Q \right)\leftrightarrow R[/latex]
Building the Truth Table
We set up a table with all possible combinations of truth values for the simple statements that make up the compound statement. Then we build new columns, one at a time. If there are parenthesis in the compound statements, we find the truth value of those first. Obviously, the best way to ensure that the intended order is followed is to use parenthesis. If there are no parenthesis the negation would come first and then the connectives and, or, if… then, if and only if.
Truth tables really become useful when analyzing more complex Boolean statements.
Example
Create a truth table for the statement [latex]A\wedge\sim\left(B\vee{C}\right)[/latex]
Try It
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