> For the complete documentation index, see [llms.txt](https://mayanktyagi3111.gitbook.io/interview-prep/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mayanktyagi3111.gitbook.io/interview-prep/graphs-bfs-and-dfs/satisfiability-of-equality-equations.md).

# Satisfiability of Equality Equations

Given an array equations of strings that represent relationships between variables, each string `equations[i]` has length `4` and takes one of two different forms: `"a==b"` or `"a!=b"`.  Here, `a` and `b` are lowercase letters (not necessarily different) that represent one-letter variable names.

Return `true` if and only if it is possible to assign integers to variable names so as to satisfy all the given equations.

**Example 1:**

```
Input: ["a==b","b!=a"]
Output: false
Explanation: 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.
```

**Example 2:**

```
Input: ["b==a","a==b"]
Output: true
Explanation: We could assign a = 1 and b = 1 to satisfy both equations.
```

**Example 3:**

```
Input: ["a==b","b==c","a==c"]
Output: true
```

**Example 4:**

```
Input: ["a==b","b!=c","c==a"]
Output: false
```

**Example 5:**

```
Input: ["c==c","b==d","x!=z"]
Output: true
```

**Note:**

1. `1 <= equations.length <= 500`
2. `equations[i].length == 4`
3. `equations[i][0]` and `equations[i][3]` are lowercase letters
4. `equations[i][1]` is either `'='` or `'!'`
5. `equations[i][2]` is `'='`

```java
class Solution {
    Map<Character, Character> parent;

    public boolean equationsPossible(String[] equations) {
        parent = new HashMap<>();
        for (String edge : equations) {
            char var1 = edge.charAt(0);
            char var2 = edge.charAt(3);
            if (!parent.containsKey(var1))
                parent.put(var1, var1);
            if (!parent.containsKey(var2))
                parent.put(var2, var2);
            if (edge.charAt(1) == '=')
                if (var1 != var2)
                    union(var1, var2);
        }
        for (String edge : equations) {
            if (edge.charAt(1) == '!') {
                char p1 = findParent(edge.charAt(0));
                char p2 = findParent(edge.charAt(3));
                if (p1 == p2)
                    return false;
            }
        }
        return true;
    }

    public void union(char v1, char v2) {
        char p1 = findParent(v1);
        char p2 = findParent(v2);
        if (p1 != p2)
            parent.put(p1, p2);
    }

    public char findParent(char c) {
        if (c != parent.get(c))
            parent.put(c, findParent(parent.get(c)));
        return parent.get(c);
    }
}
```