Definition 1.1.1. Logical Propositions.
A logical proposition or logical statement is a sentence which is either true or false, but not both.
Section 1.1 Propositional Logic
The fundamental building block of mathematics that we will be exploring in this course is logical statements/propositions. This section explores what they are and introduces operations that we do with them.
Subsection The Basics
Example 1.1.2.
Which of the following are logical propositions?
- This is a course in discrete mathematics
- Chocolate cupcakes are the best
- \(\displaystyle 1 - 3 = 4\)
- Wichita is the capitol of Kansas
- What are you doing?
- 1 + 2
Video / Answer.
Solution.
- This is a statement. It happens to be true.
- This is not a statement. Although it is my opinion that chocolate cupcakes are the best, it is not something that is true or false -- it depends.
- This is a mathematical statement. “One minus three is four.” is a false statement.
- This is a statement. It’s false, since Topeka is the capitol.
- This is a question; it’s not a statement.
- This isn’t a statement either. The mathematical operation “Add one to two” does not have a truth value, it’s just an instruction.
The last example raises an important distinction: not everything that looks like math is a logical statement. A question you might ask yourself is “what is the result?” If I wrote “1 + 2”, the result is a number. You’d say it’s 3. If I wrote “1 - 3 = 4”, you’d say, “No, that’s false!” The result is a truth value.
Definition 1.1.3. Logical Negation.
Let \(p\) be a logical proposition. The negation of \(p\text{,}\) denoted by \(\neg p\) has the opposite truth value of \(p\text{.}\)
We can negate a statement like “1- 3 = 4”. It’s negation is “\(1-3 \ne 4\)”. We can’t negate something that isn’t a statement -- Asking the opposite of “1+3” is meaningless.
Example 1.1.4.
What are the logical negations of each of the following?
- This is a course in discrete mathematics
- \(\displaystyle 1- 3 = 4\)
- Wichita is the capitol of Kansas
Definition 1.1.5. Logical Operations.
Let \(p\) and \(q\) be propositions. The conjunction of \(p\) and \(q\text{,}\) denoted \(p \wedge q\text{,}\) is the proposition “\(p\) and \(q\)”.
The disjunction of \(p\) and \(q\text{,}\) denoted \(p \vee q\text{,}\) is the proposition “\(p\) or \(q\) (or both)”.
The logical disjunction is an “inclusive or”. On the other hand, we define the “exclusive or” of \(p\) and \(q\) to be the proposition “\(p\) or \(q\) but not both”. We won’t be using it in Discrete 1, so we won’t give it a special symbol.
Definition 1.1.6. Conditional Statements.
Let \(p\) and \(q\) be propositions. The conditional statement is the compound proposition “if \(p\) then \(q\)”. The conditional is denoted by \(p \to q\text{.}\)
We call \(p\) the hypothesis or antecedent or premise, and \(q\) is the conclusion or consequence.
Example 1.1.7.
Write the following as a simple English expression, letting \(p\) be the statement “it rains” and \(q\) be the statement “I complain about the weather”.
- \(\displaystyle p\to q\)
- \(\displaystyle p \vee q\)
- \(\displaystyle q \to p\)
- \(\displaystyle \neg q \to \neg p \)
What is the logical negation of \(p \to q\) in simple English?
Note 1.1.8.
There are many ways to phrase the conditional statement \(p \to q\text{.}\) Here are just a few common ones:
- If \(p\text{,}\) then \(q\text{.}\)
- \(p\) implies \(q\text{.}\)
- \(p\) only if \(q\text{.}\)
- \(p\) if sufficient for \(q\text{.}\)
- \(q\) is necessary for \(p\text{.}\)
- \(q\) if \(p\text{.}\)
- \(q\) whenever \(p\text{.}\)
- \(q\) unless \(\neg p\text{.}\)
Definition 1.1.9. Converse/Inverse/Contrapositive.
Let \(p\) and \(q\) be propositions. For the conditional \(p \to q\text{,}\) we define:
- The converse is \(q \to p\)
- The inverse is \(\neg p \to \neg q\)
- The contrapositive is \(\neg q \to \neg p\)
Definition 1.1.10. Biconditional Statements.
Let \(p\) and \(q\) be propositions. The biconditional of \(p\) and \(q\text{,}\) is the statement “\(p\) if and only if \(q\)”, denoted \(p \leftrightarrow q\text{.}\)
Other ways to phrase an “if and only if” statement:
- \(p\) iff \(q\text{.}\)
- \(p\) is necessary and sufficient for \(q\text{.}\)
- If \(p\) then \(q\) and conversely.
Just as with arithmetic operations (\(+, -, \times, \div\)) on numbers, we need to define an order of operations so that compound propositions can be understood without grouping symbols. Though for clarity, we will generally write grouping symbols.
| Operator | Precedence |
|---|---|
| \(\neg\) | highest |
| \(\wedge, \vee \) | next, from left to right |
| \(\to, \leftrightarrow \) | lowest, left to right |
For example:
\begin{align*}
\amp\neg p \vee q \wedge r \to p \wedge q\\
\amp\equiv \left(\left(\left(\neg p\right)\vee q\right) \wedge r\right) \to \left(p \wedge q \right)
\end{align*}
Subsection Truth Tables for Logical Connectives
Truth tables allow us to uniquely determine the truth value of a compound proposition, based on the truth values of the simple statements from which it is made. Below are the truth tables for conjunction \(\wedge\text{,}\) disjunction \(\lor\text{,}\) conditional \(\to\text{,}\) biconditional \(\leftrightarrow\text{,}\) exclusive or \(\oplus\text{,}\) and negation \(\neg\text{.}\)
| \(p\) | \(q\) | \(p\wedge q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| \(p\) | \(q\) | \(p\vee q\) |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| \(p\) | \(q\) | \(p\to q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
| \(p\) | \(q\) | \(p\leftrightarrow q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
| \(p\) | \(\neg p\) |
|---|---|
| T | F |
| F | T |
Example 1.1.11.
Construct a truth table for each of the following statements:
- \(\displaystyle p \to (\neg q \vee r)\)
- \(\displaystyle p \wedge \neg q\)
- \(\displaystyle (p \to q) \wedge (\neg p \to q)\)
- \(\displaystyle (p\wedge r) \vee (q \wedge r)\)
Example 1.1.12.
Imagine your teacher makes the following (unethical) statement, “if you are a cat lover, then I’ll give you an A in this class.”
- Do you not love cats? If so, it doesn’t matter whether you get an A or not, what your teacher said was not a lie.
- Are you a cat lover? Then the only way the teacher lied to you is if you didn’t get an A.
Subsection Computer Corner
The objects that are logical propositions in mathematics are
bool Boolean datatypes in computer science. For example, the clause 5 <= 3 will evaluate to False. This corresponds to the proposition \(p:=\)“5 \(\le\) 3”\(\equiv F\text{.}\)
In C-like syntax:
- Logical and, \(p \land q\text{,}\) is in code
p && q - Logical or, \(p \lor q\text{,}\) is in code
p || q - Logical negation, \(\neg p\text{,}\) is the code
!p - The conditional is an
if...thenstatement
So a block of code such as:
if (collision == 1 && object==sword && !blocking){
// hit by a sword, take damage
health--;
}
corresponds to a logical statement of the form \((p \land q \land \neg r) \to s\text{.}\) Note that
health--; isn’t a statement. It’s an operation to decrement the health; it isn’t true or false.Exercises Exercises
1.
Construct a truth table for the compound statement \(((p \to q) \land \neg p) \to \neg q\)
Solution.
| \(p\) | \(q\) | \(((p \to q) \land \neg p) \to \neg q\) |
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |
2.
Consider the statement about a party, “If it’s your birthday or there will be cake, then there will be cake.”
- Translate the above statement into symbols. Clearly state which statement is \(P\) and which is \(Q\text{.}\)
- Make a truth table for the statement.
- Assuming the statement is true, what (if anything) can you conclude if there will be cake?
- Assuming the statement is true, what (if anything) can you conclude if there will not be cake?
- Suppose you found out that the statement was a lie. What can you conclude?
Solution.
- \(P\text{:}\) it’s your birthday; \(Q\text{:}\) there will be cake. \((P \vee Q) \to Q\)
- Hint: you should get three T’s and one F.
- Only that there will be cake.
- It’s NOT your birthday!
- It’s your birthday, but the cake is a lie.
3.
Make a truth table for the statement \((p \vee q) \to (p \wedge q)\text{.}\)
Solution.
| \(p\) | \(q\) | \((p \vee q) \to (p \wedge q)\) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
4.
Make a truth table for the statement \(\neg p \wedge (q \to p)\text{.}\) What can you conclude about \(p\) and \(q\) if you know the statement is true?
Solution.
| \(p\) | \(q\) | \(\neg p \wedge (q \to p)\) |
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
If the statement is true, then both \(p\) and \(q\) are false.
5.
Make a truth table for the statement \(\neg p \to (q \wedge r)\text{.}\)
Hint.
Like above, only now you will need 8 rows instead of just 4.
6.
State the converse, inverse, and contrapositive of each of the following conditional statements:
- If it rains today, then I will bring an umbrella.
- Whenever I drive my car, I do not use my phone.
- When I stay up too late, it’s necessary that I sleep until noon.
Solution.
- Converse: “If I bring an umbrella then it rains today.”. Inverse: “If it doesn’t rain today then I won’t bring an umbrella.” Contrapositive: “If I won’t bring an umbrella, then it isn’t raining today”.
- The conditional “Whenever I drive my car, I do not use my phone” is “If I drive my car, then I don’t use my phone.” Now find the other statements.
- The conditional “When I stay up too late, it’s necessary that I sleep until noon” is “If I stay up too late, then it’s necessary that I sleep until noon.” Now find the other statements.
7.
A classic example is that we’re on the island of knights and knaves. Knights always tell the truth. Knaves always lie.
We encounter two people A and B.
- A says: “B is a knight.”
- B says: “The two of us are opposite types.”
What is the conclusion?
Solution.
A and B are both lying knaves.
8.
This time you encounter two people, A, and B, and A says “I am a knave or B is a knight,” and B says nothing. What can you conclude?
Hint.
What happens if A is a knight and is telling the truth? What happens if A is lying? Which scenarios are impossible? What must the answer be?
9.
This is a favorite of my daughter. You encounter a guard standing at a fork in the road. It is not known whether the guard is a knight or a knave, that is, that they will (always) tell the truth or (always) lie. One of the paths leads to great treasure, the other leads to a violent and scary death. You are allowed to ask one and only one question to the guard.
What can you ask the guard in order to ensure you go on the path towards the treasure?
Hint.
The question is complicated, by not too complicated.
