05AB1E, 1 byte

Code:

_

Explanation:

_     # Logical not

Try it online!

Jelly, 1 byte

¬

Try it online!

¬ is logical NOT (1 if false-y, else 0). C (“complement”, 1−z) also works.

Haskell, 7 bytes

map not

Example:

Prelude> (map not) [False, True, True]
[True,False,False]

Matlab, 4 1 bytes

This should be self explanatory.

~

Matlab has the one-byte negation operator ~, if you want a function you can use @not.

Javascript ES6, 15 bytes

a=>a.map(b=>!b)

Not much explanation needed I guess

f=
a=>a.map(b=>!b)

a.innerHTML = `[true, false, 1, 0] => ${ f([true, false, 1, 0]) }`

<pre id=a>

MATL, 1 byte

~

Try it online!

~ is the logical not and as many functions, it can also be applied to arrays/matrices.

R, 1 byte

!

Example:

> !c(TRUE, FALSE)
[1] FALSE  TRUE

It also works with numerical input:

> !c(1, 0)
[1] FALSE  TRUE

We're not restricted to one-dimensional arrays, either. Let's make a matrix, and randomly populate it with 0s and 1s:

> mat = matrix(rbinom(16, 1, .5), ncol=4)
> mat
     [,1] [,2] [,3] [,4]
[1,]    1    1    1    1
[2,]    1    1    0    1
[3,]    0    1    0    0
[4,]    0    1    1    0

We can invert this just as easily:

> !mat
      [,1]  [,2]  [,3]  [,4]
[1,]  TRUE FALSE FALSE FALSE
[2,]  TRUE FALSE  TRUE  TRUE
[3,]  TRUE  TRUE  TRUE  TRUE
[4,] FALSE FALSE FALSE  TRUE

We can continue to do this for arbitrary numbers of dimensions. Here's an example on a four-dimensional array:

> bigarray = array(rbinom(32, 1, 0.5), dim=c(2,2,2,2))
> bigarray
, , 1, 1

     [,1] [,2]
[1,]    0    0
[2,]    0    0

, , 2, 1

     [,1] [,2]
[1,]    1    0
[2,]    0    0

, , 1, 2

     [,1] [,2]
[1,]    0    1
[2,]    0    1

, , 2, 2

     [,1] [,2]
[1,]    1    0
[2,]    1    1

> !bigarray
, , 1, 1

     [,1] [,2]
[1,] TRUE TRUE
[2,] TRUE TRUE

, , 2, 1

      [,1] [,2]
[1,] FALSE TRUE
[2,]  TRUE TRUE

, , 1, 2

     [,1]  [,2]
[1,] TRUE FALSE
[2,] TRUE FALSE

, , 2, 2

      [,1]  [,2]
[1,] FALSE  TRUE
[2,] FALSE FALSE

Doesn't work for characters, I'm afraid.

> !"Hello world"
Error in !"Hello world" : Invalid argument type.

C, 47 Bytes

f(char*s){for(;*s;putchar(*s&72?*s:*s^1),s++);}

Run using this main function

main(c,v)char**v;
{
    f(v[1]);
}

and input like this

a.exe [1,0,1,1,0]
[0,1,0,0,1]

Labyrinth, 9 bytes

,$:)%#$.,

Try it online! Assumes newline-separated input with a trailing newline. Thanks to @MartinEnder for help with golfing.

This program's a bit weird for a Labyrinth program - it doesn't make use of the 2D nature of the language, and it actually bounces back and forth. On the first forward trip, we have:

[Moving rightward]
,            Read char c of input
 $           XOR c with implicit 0 at bottom of stack
  :)%        Calculate c % (c+1), erroring out if c == -1 from EOF, otherwise returns c
     #$      XOR with (length of stack == 1)
       .     Output (c^1) as char
        ,    Read newline

[Moving leftward]
       .     Output newline
      $      XOR two implicit 0s, stack [0]
    %#       Mod with (length of stack == 1), giving stack [0]
 $:)         Increment, duplicate then XOR, stack still [0]
,            Read char c of input

The next occurence of $ then XORs the existing 0 on the stack with c, as opposed to an implicit 0 at the bottom of the stack like in the first run. Both situations leave the stack as [c], and the program repeats thereafter.

Alternative 9-bytes:

,:):/$.:,
,::)/)$.,
,:):%)$.,

J, 2 bytes

-.

This is the negation verb.

Test case

   -. 0 1 0 0 1
1 0 1 1 0

Python, 27 25 24 bytes

Thanks to Lynn for golfing off two bytes, and xnor and Mego for golfing off another.

lambda a:[b^1for b in a]

Mathematica, 7 bytes

Not/@#&

or without letters:

!#&/@#&

As for the syntactic sugar: & marks the right end of an unnamed function and has very low precedence. # refers to the first argument of the nearest and enclosing &. ! is the operator for Not. So !#& is just an unnamed function that negates its argument, in other words its identical to the built-in Not. /@ is the operator for Map. So the code would also be equivalent to the somewhat more readable Map[Not, #]&.

JAISBaL, 1 byte

!

Like all the other 1-byte answers, this is the negation operator, which can operate over an array if needed. This leaves the output on the stack, which is printed at the end of the program.

For two bytes, the array can be explicitly printed:

!§

Input is in JAISBaL's incredibly odd array format (which I did invent, but I don't like it...).

Test Cases (Output from the Java interpreter, 3.0.5):

Enter a value > [true][false]


--------------------
Stack: [[false, true]]
Locals: {}
----------------------------------------
Enter a value > [false][false][true][false][false]


--------------------
Stack: [[true, true, false, true, true]]
Locals: {}

IBM/Lotus Notes Formula, 2 bytes

!a

Usage:

Create a Notes form with two fields named a and b.

a (input) = editable, number, multi-value, comma separated

b (output) = computed, number, multi-value, comma separated

Paste the above formula into b and give a a default value of 0.

Create a new document with the form, enter a binary list in a and press F9 to update the output.

Examples:

Works because given a list as input, Notes formula will apply whatever specified action to every element in the list.

Perl, 7 bytes

Includes +2 for -lp

Give each boolean value as 0 or 1 on its own line

invert.pl
1
1
0
^D

invert.pl:

#!/us/bin/perl -lp
$_^=1

Cheddar, 10 bytes

@.map((!))

I hope I counted right as I'm writing from phone

Java, 58 bytes

void f(boolean[]a){for(boolean i:a)System.out.print(!i);}

brainfuck (58 Bytes)

-[>+<-----]>--->,[<[->->+<<]>[--<]>[>]<[-<+<+>>]<+.[-]<>,]

Try it here

Ungolfed

-[>+<-----]>---     Number 48 (stands for 0)
>,                  Read in first point
[               
    <[->->+<<]      Subtract 1 from 48 flag, subtract 1 from read data, add 1 for new flag
    >           
    [--<]           If sitting on 1 (true) subtract 2 and move left)
        >[>]<       Move to 48 flag
        [-<+<+>>]   Add 48 to data point
        <+.[-]<     Add 1 move print, zero cell, move to new 48 cell
        >,          Read in next point
]                   Loop if input remaining

Takes an input of undivided 1s or 0s (11001).

Perl 6, 4 bytes

"French"/Unicode version:

!«*

"Texas"/ASCII version:

!<<*

Input is a single value which can be treated as a list.

This is a a Whatever lambda (*) with the logical not prefix operator (!) combined using prefix hyper operator («).

Effectively this is the same as:

-> $_ { $_.values.hyper.map: &prefix:<!> }
# ( currently the Rakudo implementation doesn't actually do the ｢.hyper｣ call,
#   but prefix ｢«｣ is specifically designated for doing things in parallel )

Usage:

# pretend it's a method
say (True,False,True,True).&( !«* );
# (False True False False)

say ( !«* )( (False,False,True,False,False) );
# (True True False True True)


# give it a lexical name
my &list-invert = !«*;

#              v¯¯ a space is necessary here
say list-invert (True,False);
# (False True)

say (False,True).&list-invert;
# (True False)

C#, 19 bytes

as an annonymous function, takes a bool[] and returns an IEnumerable

b=>b.Select(x=>!x);

or in 36 bytes with

dynamic f(bool[]b)=>b.Select(x=>!x);

CJam, 4 bytes

{:!}

Input is a list of 0s and 1s.

Try it online!

Brachylog, 7 bytes

:{-$_}a

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Explanation

:{   }a   Apply this predicate to each element of the Input
  -         Decrement
   $_       Negate

PowerShell, 15 bytes

$args[0]|%{!$_}

I think this may even work in v1, hence I left the version number off the title. Loops through the input $args and negates each item in turn. That resulting array is left on the pipeline.

The neat thing, however, is because PowerShell is so loose on its casting requirements, you can do a completely mixed input and get an appropriate Boolean output. For example, here are the literal Boolean values $false/$true, the numbers 0 1 and 123456789 as integers, an empty string, a non-empty string, an empty array, and a non-empty array --

PS C:\Tools\Scripts\golfing> .\invert-a-boolean-array.ps1 @($false,$true,0,1,123456789,'','foo',@(),@(1,1))
True
False
True
False
False
True
False
True
False

Japt, 3 bytes

¡!X

Japt doesn't have boolean input, so input is an array of 0s and 1s. Test it online!

How it works

¡    // Map each item X in the input to
 !X  //  the boolean NOT of X.
     // Implicit output

Retina, 3 bytes

%`0

Try it online!

For each line (%), count the number of 0s.

Python 2, 24 bytes (non-competing)

lambda a:[i-1for i in a]

Logic is similar to Steven's, but I tried to use this comment's idea, but different, because it still takes 0/1 arrays, not 0/-1. There is no byte shaving for using 0/-1, so let's be sane. Note that this is non-competing, until Steven or Lynn allows me to use the idea. If so, I might remove the non-competing mark. Note that this code cannot be shamelessly stolen, it's still here. Only Steven can use it for his answer.

Perl 6, 9 bytes

*.map: !*

Usage:

say (*.map: !*)((0, 1, 1)) # (True False False)

Logicode, 10 bytes

out !input

Simple, really.

Takes input as a binary string, as Logicode doesn't have support for lists (so [true, false] would be 10).

The out outputs the line's result.

The ! command calculates the NOT of every bit in the string, so something like !111 would be 000.

The input is input.

Clojure, 12 bytes

#(map not %) 

Basically the same as the Haskell answer. Unfortunately, Clojure doesn't have implicit partial application; thus the function macro.

Swift 3 (7 bytes)

.map(!)

e.g.

[true, false].map(!)

Explanation

Seems pretty obvious. Calls map on the array [true, false]. The one "gotcha" is that, in Swift, operators are just functions and can be passed around as arguments. This means map(!) is passing the "not" function ! into map.