What is the JavaScript equivalent of jQuery's hide() and show()?

this is simple

document.getElementById('myElement').style.display = 'block'; // show
document.getElementById('myElement').style.display = 'none'; // hide

add a onSelect="selectOptionsupdated(this) in your select

then

function selectOptionsupdated(select){
//do your stuff here  
}

var myButton = document.getElementById('myButton');

//hide
myButton.style.display = 'none';

//show
myButton.style.display = 'block';

Update for your select tag..try this

html

<select id="list">
<option id="1">Science</option>
</select>

js

var list = document.getElementById('select');

list.addEventListener('change', listSelect, false);

function listSelect(){    
    var selected = list.options[list.selectedIndex].value;//Selected option value    //hide
    myButton.style.display = 'none';

    //show
    myButton.style.display = 'block'; 
}