[Vuejs]-Vue.js not updating automatically

here is your code working and I updated your fiddle
Adding the this statement and adding the event param inside optionButtonClicked

const ProgrammingLanguages = new Vue({
            el:".pl_wrapper",
            data:{
                name: "aa",
                show1: false,
                show2: false
            },
            methods: {
                optionButtonClicked(event){
                    var indexOfClicked  = index(event.target,event.target.className);
                    this.show1 = (indexOfClicked==0) ? true : false
                    this.show2 = (indexOfClicked==1) ? true : false
          console.log(this.show2)
                }
            }
        });
        function index(element,className){
        var allElements = document.getElementsByClassName(className);
        for (var i = 0; i < allElements.length; i++) {
            if(allElements[i]==element){
                return i;
            }
        }
    }