PHP - Checkbox saving value to Database

I have a few checkbox in a form and I'm setting each value if its checked or unchecked. So a checkbox that is checked has a value of TRUE and unchecked as FALSE If I have all of the checkbox checked(value is TRUE) and I unchecked one of them, the other checkbox also unchecked and the value on the database also become FALSE. I can't understand the behavior below is my code

HTML

                    <label><input type="checkbox" id="trans_val_identical" value="FALSE">II Transaction Value of Identical Goods</label>

                   <label><input type="checkbox" id="trans_val_identical" checked="checked" value="FALSE">II Transaction Value of Identical Goods</label>


              </div>
              <div class="checkbox">

                     <label><input type="checkbox" id="trans_val_similar" value="FALSE">III Transaction Value of Similar Goods</label>

                     <label><input type="checkbox" id="trans_val_similar" checked="checked" value="FALSE">III Transaction Value of Similar Goods</label>

              </div>
              <div class="checkbox">

                     <label><input type="checkbox" id="deductive_val" value="FALSE">IV Deductive Value</label>

                     <label><input type="checkbox" id="deductive_val" checked="checked" value="FALSE">IV Deductive Value</label>

              </div>
              <div class="checkbox">

                     <label><input type="checkbox" id="computed_val" value="FALSE">V Computed Value</label>


                     <label><input type="checkbox"  id="computed_val" checked="checked" value="FALSE">V Computed Value</label>

              </div>
              <div class="checkbox">

                     <label><input type="checkbox" id="fallback_val" value="FALSE">VI Fallback Value</label>

                     <label><input type="checkbox" id="fallback_val" checked="checked" value="FALSE">VI Fallback Value</label>

JS

    $('#trans_val_identical').change(function(){
     if($(this).attr('checked')){
          $(this).val('TRUE');
     }else{
          $(this).val('FALSE');
     }
});
$('#trans_val_similar').change(function(){
     if($(this).attr('checked')){
          $(this).val('TRUE');
     }else{
          $(this).val('FALSE');
     }
});
$('#deductive_val').change(function(){
     if($(this).attr('checked')){
          $(this).val('TRUE');
     }else{
          $(this).val('FALSE');
     }
});
$('#computed_val').change(function(){
     if($(this).attr('checked')){
          $(this).val('TRUE');
     }else{
          $(this).val('FALSE');
     }
});
$('#fallback_val').change(function(){
     if($(this).attr('checked')){
          $(this).val('TRUE');
     }else{
          $(this).val('FALSE');
     }
});