Boolean algebra is a way of describing logic in a mathematical way. In boolean algebra you work with two states – true and false – rather than numbers. 1 is used to denote true and 0 is used to denote false. The three most basic operations in boolean algebra are **and, or, **and** not.** These operations can be chained together to determine the truth of a statement.

**The And Operator**

The **and** operator which sometimes uses the symbol **∧** is the first operator that we will be using. The following is a truth table that can be used to determine the result of **and**ing two values together.

Operation | Result |

true ∧ true | true |

true ∧ false | false |

false ∧ true | false |

false ∧ false | false |

This should give you a better idea of how logical operators work. The **and** operator will take two boolean values and will give back a single boolean value. If you look at the truth table for **and** you can see that **and** will only be true when it is given two true values. Here is a longer example using **and**s:

(true ∧ false) ∧ (true ∧ true) (false) ∧ (true) false

Super simple so far! Let’s look at the next operator.

**The Or Operator**

Similar to the **and **operator, the **or** operator takes two boolean values and returns a single boolean value. The symbol for the **or** operator is **∨**. Let’s take a look at the truth table:

Operation | Result |

true ∨ true | true |

true ∨ false | true |

false ∨ true | true |

false ∨ false | false |

So the **or** operator is sort of the opposite of the **and** operator. The **and** operator is only true when both values are true but the **or** operator is only false when both values are false. Let’s take a look at the same example from earlier using **or**s instead.

(true ∨ false) ∨ (true ∨ true) (true) ∨ (true) true

Very similar to **and**. The next operator will work a little differently but is probably the most intuitive.

If you are getting confused between the **∨** and **∧** operators, here is a little trick I use to help myself remember:

**Negation with the Not Operator**

The **not** operator (which uses the **!** or **¬** symbol) is a little bit different because it affects a single value rather than operating on multiple values. Here is the truth table:

Operation | Result |

¬(true) | false |

¬(false) | true |

All the **not** operator does is flip the value of a statement! Now let’s try some equations using all three of the operators.

A = true B = false (¬A V B) Λ (A V B) (false V false) Λ (true V false) false Λ true false

Now that we have seen how these basic operators work, let’s look at how they can be applied when writing code.

**Applications in Computer Programming**

Let’s imagine that we have a program that is performing some kind of user access functionality. Take a look at the psuedocode below.

bool authenticated = authenticateUser(username, password); string userType = getUserType(username); if(authenticated && userType = "admin") { printf("Welcome admin! \n"); } else if (authenticated && userType != "admin") { printf("Welcome %s \n", getUserName(username)); } else { printf("Login failed. Please try again"); }

Control flow for the program is being managed by the same type of logical equations that we just solved. You’ll notice that I am using the && (logical **and** operator) – the C programming language equivalent of Λ. If the equation evaluates to true then we enter the code block. If it does not, then we skip to the next block and do any evaluations there. Here we can see two logical equations:

authenticated && userType = "admin" authenticated && userType != "admin"

Let’s convert these to the symbolic logic notation from earlier in this post.

authenticated = true (userType = "admin") = false true && false true Λ false false

Since the first equation evaluated to false, we now have to evaluate the second equation.

authenticated = true (userType = "admin") != true true && true true Λ true true

Since this equation evaluated to true will execute the second block of code printing “Welcome” and the users username.

In the next tutorial I will cover more logical operators.