Boolean Truth Tables

In my Computer Science class, we are revisiting Boolean operators and are looking more in-depth at Boolean Truth Tables.

In this post, I will look at the Boolean operators of AND, OR, NAND, NOR, and XOR.

For clarification, the following are considered equal and will be used in the post:

a AND b > c is equal to a . b > c

a NAND b > c is equal to a . b > c

a OR b > c is equal to a + b > c

a NOR b > c is equal to a + b > c

a XOR b > c is equal to a ⊕ b > c

For additional clarification, here are the logic gate representations of each of the objects that are presented:

Boolean AND & NAND Logic Gates
Boolean AND & NAND Logic Gates
Boolean OR & NOR Logic Gates
Boolean OR & NOR Logic Gates
Boolean XOR & NOT Logic Gates
Boolean XOR & NOT Logic Gates

So, let’s build our truth tables for each of the above named scenarios:

a . b > c
A B Result
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE

In this scenario, let’s say that a = 10, b = 5, and c = 7. In this case, the statement A > C is TRUE. However, the statement B > C is FALSE. If we look at the truth table above, the result is that the entire statement is now FALSE.

For an AND statement to be TRUE, all parts of the statement must be TRUE.

a + b > c
A B Result
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

In this scenario, let’s say that a = 10, b = 5, and c = 7. In this case, the statement A > C is TRUE. However, the statement B > C is FALSE. If we look at the truth table above, the result is that the entire statement is now TRUE.

For an OR statement to be TRUE, at least one part up to all parts of the statement must be TRUE.

a . b > c
A B Result
TRUE TRUE FALSE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE TRUE

In this scenario, let’s say that a = 10, b = 5, and c = 7. In this case, the statement A > C is TRUE. However, the statement B > C is FALSE. If we look at the truth table above, the result is that the entire statement is now TRUE.

For a NAND statement to be TRUE, at least one part up to all parts of the statement must be FALSE.

a + b > c
A B Result
TRUE TRUE FALSE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE TRUE

In this scenario, let’s say that a = 10, b = 5, and c = 7. In this case, the statement A > C is TRUE. However, the statement B > C is FALSE. If we look at the truth table above, the result is that the entire statement is now FALSE.

For a NOR statement to be TRUE, all parts of the statement must be FALSE.

a ⊕ b > c
A B Result
TRUE TRUE FALSE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

In this scenario, let’s say that a = 10, b = 5, and c = 7. In this case, the statement A > C is TRUE. However, the statement B > C is FALSE. If we look at the truth table above, the result is that the entire statement is now TRUE.

For a XOR statement to be TRUE, at least one condition in the statement must be TRUE. However, if all conditions in the statement are TRUE, then the result is FALSE.