<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 Solution {
public:
    vector<int> parent;
    vector<int> rank;
    int find(int p){
        if(p == parent[p]){
            return p;
        }
        return parent[p] = find(parent[p]);
    }
    void Union(int x , int y){
        int x_p = find(x);
        int y_p = find(y);
        if(x_p == y_p) return;
        else if(rank[x_p] > rank[y_p]) parent[y_p] = x_p;
        else if(rank[x_p]<rank[y_p]) parent[x_p] = y_p;
        else {
            parent[x_p] = y_p;
            rank[y_p]++;
        }
    }
    bool equationsPossible(vector<string>& equations) {
        parent.resize(26);
        rank.resize(26 , 0);
        for(int i = 0 ; i < 26 ; i++) parent[i] = i;
        for(auto &p : equations){
            if(p[1] == '=') {
                Union(p[0] - 'a' , p[3] - 'a');
            }
        }
        for(auto &p : equations){
            if(p[1] == '!'){
                if(find(p[0] - 'a') == find(p[3] - 'a')) return false;
            }
        }
        return true;
    }
};
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