<p>You are given an array of strings <code>equations</code> that represent relationships between variables where each string <code>equations[i]</code> is of length <code>4</code> and takes one of two different forms: <code>&quot;x<sub>i</sub>==y<sub>i</sub>&quot;</code> or <code>&quot;x<sub>i</sub>!=y<sub>i</sub>&quot;</code>.Here, <code>x<sub>i</sub></code> and <code>y<sub>i</sub></code> are lowercase letters (not necessarily different) that represent one-letter variable names.</p>

<p>Return <code>true</code><em> if it is possible to assign integers to variable names so as to satisfy all the given equations, or </em><code>false</code><em> otherwise</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> equations = [&quot;a==b&quot;,&quot;b!=a&quot;]
<strong>Output:</strong> false
<strong>Explanation:</strong> If we assign say, a = 1 and b = 1, then the first equation is satisfied, but not the second.
There is no way to assign the variables to satisfy both equations.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> equations = [&quot;b==a&quot;,&quot;a==b&quot;]
<strong>Output:</strong> true
<strong>Explanation:</strong> We could assign a = 1 and b = 1 to satisfy both equations.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= equations.length &lt;= 500</code></li>
	<li><code>equations[i].length == 4</code></li>
	<li><code>equations[i][0]</code> is a lowercase letter.</li>
	<li><code>equations[i][1]</code> is either <code>&#39;=&#39;</code> or <code>&#39;!&#39;</code>.</li>
	<li><code>equations[i][2]</code> is <code>&#39;=&#39;</code>.</li>
	<li><code>equations[i][3]</code> is a lowercase letter.</li>
</ul>

class disjointSet{
    vector<int> par,size; 
    public:
    disjointSet(int n){
        par.resize(n+1,0);
        size.resize(n+1,0);
        for(int i=0;i<n;i++){
            par[i]=i;
        }
    }
    int find(int u){
        if(u==par[u]){
            return u;
        }
        return par[u]=find(par[u]);
    }
    void unionBysize(int u,int v){
        int Uu=find(u);
        int Uv=find(v);
        if(Uu==Uv){
            return ;
        }
        if(size[Uu]>size[Uv]){
            par[Uv]=Uu;
            size[Uu]+=size[Uv];
        }else{
            par[Uu]=Uv;
            size[Uv]+=size[Uu];
        }
    }
};

class Solution {
public:
    bool equationsPossible(vector<string>& equations) {
       
        int n=equations.size();
        map<int,int> mp;
        int k=0;
        for(auto it:equations){
           
            if(mp.find(it[0]-'a')==mp.end()){
            mp[it[0]-'a']=k;
            k++;
            }
             if(mp.find(it[3]-'a')==mp.end()){
            mp[it[3]-'a']=k;
            k++;
            }
        }
       
        disjointSet ds(mp.size());
        for(auto equation:equations){
             if(equation[1]=='='){
            ds.unionBysize(mp[equation[0]-'a'],mp[equation[3]-'a']);
             }
        }
        for(int i=0;i<n;i++){
            if(equations[i][1]=='!'){
                 if(ds.find(mp[equations[i][0]-'a'])==ds.find(mp[equations[i][3]-'a'])){
                return false;
                }
            }
             if(equations[i][1]=='='){
                 if(ds.find(mp[equations[i][0]-'a'])!=ds.find(mp[equations[i][3]-'a'])){
                return false;
                }
            }
           
        }
        return true;
        

    }
};
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