An overview of the Logical Math tools
The Logical Math tools evaluate the values of the inputs and determine the output values based on Boolean logic. The tools are grouped into four main categories: Boolean, Combinatorial, Logical, and Relational.
The following is a list of all the tools within each grouping:
 Boolean:
Boolean And, Boolean Or, Boolean XOr, Boolean Not
 Combinatorial:
Combinatorial And, Combinatorial Or, Combinatorial XOr
 Relational:
Equal To, Not Equal, Greater Than, Greater Than Equal, Less Than, Less Than Equal
 Logical:
Diff, InList, Is Null, Over, Test
Boolean logic generally applies a binary condition to the inputs and evaluates to a binary condition for the output. The binary condition can be expressed in several ways: "1" and "0", "True" and "False", "yes" and "no", "on" and "off", and so forth. In the Logical Math tools, the False condition is represented with a value of 0, and the True condition as any value other than 0.
The Boolean tools evaluate the inputs only as True or False conditions and return the result of the particular tool as a 1 or 0 (True or False) Boolean value. The Combinatorial tools identify unique combinations of input values based on the logic of the particular tool and return a different value for each unique combination. The Relational tools compare the values of one input relative to another and return the result of the particular tool as 1 or 0 Boolean value. The Logical tools have different ways to apply Boolean logic, such as identifying only the input cells that are NoData, or using a logical expression that you define to determine which cells are evaluated as true.
The following tables list the available Logical Math tools by category and provide a description of each:
Boolean evaluation
Tool 
Description 

Performs a Boolean And operation on the cell values of two input rasters. If both input values are true (nonzero), the output value is 1. If one or both inputs are false (zero), the output is 0. 

Performs a Boolean Not (complement) operation on the cell values of the input raster. If the input values are true (nonzero), the output value is 1. If the input values are false (zero), the output is 0. 

Performs a Boolean Or operation on the cell values of two input rasters. If one or both input values are true (nonzero), the output value is 1. If both input values are false (zero), the output is 0. 

Performs a Boolean eXclusive Or operation on the cell values of two input rasters. If one input value is true (nonzero) and the other false (zero), the output is 1. If both input values are true or both are false, the output is 0. 
Combinatorial evaluation
Tool 
Description 

Performs a Combinatorial And operation on the cell values of two input rasters. If both input values are true (nonzero), the output is a different value for each unique combination of input values. If one or both inputs are false (zero), the output value is 0. 

Performs a Combinatorial Or operation on the cell values of two input rasters. If either input value is true (nonzero), the output is a different value for each unique combination of input values. If both inputs are false (zero), the output value is 0. 

Performs a Combinatorial eXclusive Or operation on the cell values of two input rasters. If one input value is true (nonzero) and the other false (zero), the output is a different value for each unique combination of input values. If both inputs are true or both are false, the output value is 0. 
Relational evaluation
Tool 
Description 

Performs a Relational equalto operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster equals the second raster and 0 for cells where it does not. 

Performs a Relational greaterthan operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster is greater than the second raster and 0 for cells if it is not. 

Performs a Relational greaterthanorequalto operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster is greater than or equal to the second raster and 0 if it is not. 

Performs a Relational lessthan operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster is less than the second raster and 0 if it is not. 

Performs a Relational lessthanorequalto operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster is less than or equal to the second raster and 0 where it is not. 

Performs a Relational notequalto operation on two inputs on a cellbycell basis. Returns 1 for cells where the first raster is not equal to the second raster and 0 for cells where it is equal. 
Logical evaluation
Tool 
Description 

Determines which values from the first input are logically different from the values of the second input on a cellbycell basis. If the values on the two inputs are different, the value on the first input is output. If the values on the two inputs are the same, the output is 0. 

Determines which values from the first input are contained in a set of other inputs, on a cellbycell basis. For each cell, if the value of the first input raster is found in any of the list of other inputs, that value will be assigned to the output raster. If it is not found, the output cell will be NoData. 

Determines which values from the input raster are NoData on a cellbycell basis. Returns a value of 1 if the input value is NoData and 0 for cells that are not. 

For the cell values in the first input that are not 0, the output value will be that of the first input. Where the cell values are 0, the output will be that of the second input raster. 

Performs a Boolean evaluation of the input raster using a logical expression. When the expression evaluates to true, the output cell value is 1. If the expression is false, the output cell value is 0. 