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You can filter the elements of an array by applying one or more
conditions to the array. For instance, if you want to examine only the even elements in a
matrix, find the location of all 0s in a multidimensional array, or replace
`NaN`

values in a discrete set of data. You can perform these tasks
using a combination of the relational and logical operators. The relational operators
(`>, <, >=, <=, ==, ~=`

) impose conditions on the array,
and you can apply multiple conditions by connecting them with the logical operators
`and`

, `or`

, and `not`

, respectively
denoted by `&`

, `|`

, and
`~`

.

To apply a single condition, start by creating a 5-by-5 matrix, `A`

,
that contains random integers between 1 and 15.

```
rng default
A = randi(15,5)
```

A = 13 2 3 3 10 14 5 15 7 1 2 9 15 14 13 14 15 8 12 15 10 15 13 15 11

Use the relational *less than* operator, `<`

,
to determine which elements of `A`

are less than 9. Store the result in
`B`

.

B = A < 9

B = 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0

The result is a logical matrix. Each value in `B`

represents a
logical `1`

(`true`

) or logical `0`

(`false`

) state to indicate whether the corresponding element of
`A`

fulfills the condition `A < 9`

. For example,
`A(1,1)`

is `13`

, so `B(1,1)`

is
logical `0`

(`false`

). However,
`A(1,2)`

is `2`

, so `B(1,2)`

is
logical `1`

(`true`

).

Although `B`

contains information about *which*
elements in `A`

are less than 9, it doesn’t tell you what their
*values* are. Rather than comparing the two matrices element by
element, use `B`

to index into `A`

.

A(B)

ans = 2 2 5 3 8 3 7 1

The result is a column vector of the elements in `A`

that are less
than 9. Since `B`

is a logical matrix, this operation is called
*logical indexing*. In this case, the logical array being used as an
index is the same size as the other array, but this is not a requirement. For more
information, see Using Logicals in Array Indexing.

Some problems require information about the *locations* of the
array elements that meet a condition rather than their actual values. In this example, use
the `find`

function to locate all of the elements in `A`

less than 9.

I = find(A < 9)

I = 3 6 7 11 14 16 17 22

The result is a column vector of linear indices. Each index describes the location of
an element in `A`

that is less than 9, so in practice
`A(I)`

returns the same result as `A(B)`

. The
difference is that `A(B)`

uses logical indexing, whereas
`A(I)`

uses linear indexing.

You can use the logical `and`

, `or`

, and
`not`

operators to apply any number of conditions to an array; the
number of conditions is not limited to one or two.

First, use the logical `and`

operator, denoted
`&`

, to specify two conditions: the elements must be less than 9
AND greater than 2. Specify the conditions as a logical index to view the elements that
satisfy both conditions.

A(A<9 & A>2)

ans = 5 3 8 3 7

The result is a list of the elements in `A`

that satisfy both
conditions. Be sure to specify each condition with a separate statement connected by a
logical operator. For example, you cannot specify the conditions above by
`A(2<A<9)`

, since it evaluates to ```
A(2<A |
A<9)
```

.

Next, find the elements in `A`

that are less than 9 AND even
numbered.

A(A<9 & ~mod(A,2))

ans = 2 2 8

The result is a list of all even elements in `A`

that are less than
9. The use of the logical NOT operator, `~`

, converts the matrix
`mod(A,2)`

into a logical matrix, with a value of logical
`1`

(`true`

) located where an element is evenly
divisible by 2.

Finally, find the elements in `A`

that are less than 9 AND even
numbered AND not equal to 2.

A(A<9 & ~mod(A,2) & A~=2)

ans = 8

The result, 8, is even, less than 9, and not equal to 2. It is the only element in
`A`

that satisfies all three conditions.

Use the `find`

function to get the index of the `8`

element that satisfies the conditions.

find(A<9 & ~mod(A,2) & A~=2)

ans = 14

The result indicates that `A(14) = 8`

.

Sometimes it is useful to simultaneously change the values of several existing array elements. Use logical indexing with a simple assignment statement to replace the values in an array that meet a condition.

Replace all values in `A`

that are greater than 10 with the number
10.

A(A>10) = 10

A = 10 2 3 3 10 10 5 10 7 1 2 9 10 10 10 10 10 8 10 10 10 10 10 10 10

`A`

now has a maximum value of `10`

.

Replace all values in `A`

that are not equal to 10 with a
`NaN`

value.

A(A~=10) = NaN

A = 10 NaN NaN NaN 10 10 NaN 10 NaN NaN NaN NaN 10 10 10 10 10 NaN 10 10 10 10 10 10 10

The resulting matrix has element values of 10 or `NaN`

.

Replace all of the `NaN`

values in `A`

with zeros
and apply the logical NOT operator, `~A`

.

A(isnan(A)) = 0; C = ~A

C = 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0

The resulting matrix has values of logical `1`

(`true`

) in place of the `NaN`

values, and logical
`0`

(`false`

) in place of the 10s. The logical NOT
operation, `~A`

, converts the numeric array into a logical array such
that `A&C`

returns a matrix of logical `0`

(`false`

) values and `A|C`

returns a matrix of logical
`1`

(`true`

) values.

`Logical Operators: Short Circuit`

| `and`

| `find`

| `isnan`

| `nan`

| `not`

| `or`

| `xor`

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