[jquery] Setting the value of checkbox to true or false with jQuery

Try this:

HTML:

<input type="checkbox" value="FALSE" />

jQ:

$("input[type='checkbox']").on('change', function(){
  $(this).val(this.checked ? "TRUE" : "FALSE");
})

jsfiddle

Please bear in mind that unchecked checkbox will not be submitted in regular form, and you should use hidden filed in order to do it.