I am using AJAX with JQUERY to call a PHP script to validate a user email. But, for some reason, the form submits even when it shouldn't. What am I doing wrong? I know the error is for sure not in my PHP.

My Code:

$("#signup").submit(function() {
    
var error= false;
    
var dataString = $(this).serialize();
var email= $("#email").val().trim();

        
if (email != 0) {
// Run AJAX email validation and check to see if the email is already taken
    $.ajax({  
    type: "POST",  
    url: "checkemail.php",  
    data: dataString,
        asych: false,
    success: function(data) {
        var error= false;
                    
        if (data == 'invalid') {
        var invalid= 1;
        }
        else if (data == 'taken') {
        var taken= 1;
        }
        if (invalid == 1) {
        alert('invalid email');
            error = true;
        }
        if (taken == 1) {
        alert('email taken');
        error = true;
        }
        if (error == true) {
        return false;
        }
    }
    });
}
        
    });

I am using AJAX with JQUERY to call a PHP script to validate a user email. But, for some reason, the form submits even when it shouldn't. What am I doing wrong? I know the error is for sure not in my PHP.

My Code:

$("#signup").submit(function() {
    
var error= false;
    
var dataString = $(this).serialize();
var email= $("#email").val().trim();

        
if (email != 0) {
// Run AJAX email validation and check to see if the email is already taken
    $.ajax({  
    type: "POST",  
    url: "checkemail.php",  
    data: dataString,
        async: false,
    success: function(data) {
        var error= false;
                    
        if (data == 'invalid') {
        var invalid= 1;
        }
        else if (data == 'taken') {
        var taken= 1;
        }
        if (invalid == 1) {
        alert('invalid email');
            error = true;
        }
        if (taken == 1) {
        alert('email taken');
        error = true;
        }
        if (error == true) {
        return false;
        }
    }
    });
}
        
    });