# Check if a pair of integers from two ranges exists such that their Bitwise XOR exceeds both the ranges

Given two integers A and B, the task is to check if there exists two integers P and Q over the range [1, A] and [1, B] respectively such that Bitwise XOR of P and Q is greater than A and B. If found to be true, then print “Yes”. Otherwise, print “No”.

Examples:

Input: X = 2, Y = 2
Output: Yes
Explanation:
By choosing the value of P and Q as 1 and 2 respectively, gives the Bitwise XOR of P and Q as 1^2 = 3 which is greater than Bitwise XOR of A and B A ^ B = 0.
Therefore, print Yes.

Input: X = 2, Y = 4
Output: No

Naive Approach: The simplest approach to solve the given problem is to generate all possible pairs of (P, Q) by traversing all integers from 1 to X and 1 to Y and check if there exists a pair such that their Bitwise XOR is greater than Bitwise XOR of X and Y, then print “Yes”. Otherwise, print “No”.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach`   `#include ` `using` `namespace` `std;`   `// Function to check if there exists any pair (P, Q) whose` `// Bitwise XOR is greater than the Bitwise XOR of X and Y` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    ``// Stores the Bitwise XOR of X & Y` `    ``int` `playerA = (X ^ Y);` `    ``bool` `flag = ``false``;` `    ``// Traverse all possible pairs` `    ``for` `(``int` `i = 1; i <= X; i++) {` `        ``for` `(``int` `j = 1; j <= Y; j++) {` `            ``int` `val = (i ^ j);` `            ``// If a pair exists` `            ``if` `(val > playerA) {` `                ``flag = ``true``;` `                ``break``;` `            ``}` `        ``}` `        ``if` `(flag)` `            ``break``;` `    ``}` `    ``// If a pair is found` `    ``if` `(flag)` `        ``cout << ``"Yes"``;` `    ``else` `        ``cout << ``"No"``;` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `A = 2, B = 4;` `    ``findWinner(A, B);` `    ``return` `0;` `}`

## C

 `// C program for the above approach` `#include ` `#include `   `// Function to check if there exists any pair (P, Q) whose` `// Bitwise XOR is greater than the Bitwise XOR of X and Y` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    ``// Stores the Bitwise XOR of X & Y` `    ``int` `playerA = (X ^ Y);` `    ``bool` `flag = ``false``;` `    ``// Traverse all possible pairs` `    ``for` `(``int` `i = 1; i <= X; i++) {` `        ``for` `(``int` `j = 1; j <= Y; j++) {` `            ``int` `val = (i ^ j);` `            ``// If a pair exists` `            ``if` `(val > playerA) {` `                ``flag = ``true``;` `                ``break``;` `            ``}` `        ``}` `        ``if` `(flag)` `            ``break``;` `    ``}` `    ``// If a pair is found` `    ``if` `(flag)` `        ``printf``(``"Yes"``);` `    ``else` `        ``printf``(``"No"``);` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `A = 2, B = 4;` `    ``findWinner(A, B);` `    ``return` `0;` `}`   `// This code is contributed by Sania Kumari Gupta`

## Java

 `// Java program for the above approach` `import` `java.lang.*;` `import` `java.util.*;`   `class` `GFG{` `    `  `// Function to check if there exists` `// any pair (P, Q) whose Bitwise XOR` `// is greater than the Bitwise XOR` `// of X and Y` `static` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    `  `    ``// Stores the Bitwise XOR of X & Y` `    ``int` `playerA = (X ^ Y);`   `    ``boolean` `flag = ``false``;`   `    ``// Traverse all possible pairs` `    ``for``(``int` `i = ``1``; i <= X; i++) ` `    ``{` `        ``for``(``int` `j = ``1``; j <= Y; j++)` `        ``{` `            ``int` `val = (i ^ j);`   `            ``// If a pair exists` `            ``if` `(val > playerA) ` `            ``{` `                ``flag = ``true``;` `                ``break``;` `            ``}` `        ``}`   `        ``if` `(flag)` `        ``{` `            ``break``;` `        ``}` `    ``}`   `    ``// If a pair is found` `    ``if` `(flag) ` `    ``{` `        ``System.out.println(``"Yes"``);` `    ``}` `    ``else` `    ``{` `        ``System.out.println(``"No"``);` `    ``}` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{ ` `    ``int` `A = ``2``, B = ``4``;` `    `  `    ``findWinner(A, B);` `}` `}`   `// This code is contributed by offbeat`

## Python3

 `# Python3 program for the above approach`   `# Function to check if there exists` `# any pair (P, Q) whose Bitwise XOR` `# is greater than the Bitwise XOR` `# of X and Y` `def` `findWinner(X, Y):` `    `  `    ``# Stores the Bitwise XOR of X & Y` `    ``playerA ``=` `(X ^ Y)`   `    ``flag ``=` `False`   `    ``# Traverse all possible pairs` `    ``for` `i ``in` `range``(``1``, X ``+` `1``, ``1``):` `        ``for` `j ``in` `range``(``1``, Y ``+` `1``, ``1``):` `            ``val ``=` `(i ^ j)`   `            ``# If a pair exists` `            ``if` `(val > playerA):` `                ``flag ``=` `True` `                ``break`   `        ``if` `(flag):` `            ``break`   `    ``# If a pair is found` `    ``if` `(flag):` `        ``print``(``"Yes"``)` `    ``else``:` `        ``print``(``"No"``)`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    `  `    ``A ``=` `2` `    ``B ``=` `4` `    `  `    ``findWinner(A, B)`   `# This code is contributed by bgangwar59`

## C#

 `// C# program for the above approach` `using` `System.Collections.Generic; ` `using` `System; `   `class` `GFG{` `    `  `// Function to check if there exists` `// any pair (P, Q) whose Bitwise XOR` `// is greater than the Bitwise XOR` `// of X and Y` `static` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    `  `    ``// Stores the Bitwise XOR of X & Y` `    ``int` `playerA = (X ^ Y);`   `    ``bool` `flag = ``false``;`   `    ``// Traverse all possible pairs` `    ``for``(``int` `i = 1; i <= X; i++) ` `    ``{` `        ``for``(``int` `j = 1; j <= Y; j++)` `        ``{` `            ``int` `val = (i ^ j);`   `            ``// If a pair exists` `            ``if` `(val > playerA) ` `            ``{` `                ``flag = ``true``;` `                ``break``;` `            ``}` `        ``}`   `        ``if` `(flag)` `        ``{` `            ``break``;` `        ``}` `    ``}`   `    ``// If a pair is found` `    ``if` `(flag) ` `    ``{` `        ``Console.WriteLine(``"Yes"``);` `    ``}` `    ``else` `    ``{` `        ``Console.WriteLine(``"No"``);` `    ``}` `}`   `// Driver code` `public` `static` `void` `Main(String[] args)` `{ ` `    ``int` `A = 2, B = 4;` `    `  `    ``findWinner(A, B);` `}` `}`   `// This code is contributed by amreshkumar3`

## Javascript

 ``

Output:

`No`

Time Complexity: O(X * Y)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized based on the following observations:

• For any two integers P and Q, the maximum Bitwise XOR value is (P + Q) which can only be found when there are no common bits between P and Q in their binary representation.
• There are two cases:
• Case 1: If player A has two integers that produce the maximum Bitwise XOR value, then print “No”.
• Case 2: In this case, there must have some common bit between A and B such that there always exist two integers P and Q whose Bitwise XOR is always greater than the Bitwise XOR of A and B, where (P ^ Q) = (X | Y).

Therefore, from the above observations, the idea is to check if the value of given A^B is equal to A + B or not. If found to be true, then print “No”. Otherwise, print “Yes”.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach`   `#include ` `using` `namespace` `std;`   `// Function to check if there exists` `// any pair (P, Q) whose Bitwise XOR` `// is greater than the Bitwise XOR` `// of X and Y` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    ``int` `first = (X ^ Y);` `    ``int` `second = (X + Y);`   `    ``// Check for the invalid condition` `    ``if` `(first == second) {` `        ``cout << ``"No"``;` `    ``}`   `    ``// Otherwise,` `    ``else` `{` `        ``cout << ``"Yes"``;` `    ``}` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `A = 2, B = 4;` `    ``findWinner(A, B);`   `    ``return` `0;` `}`

## Java

 `// Java program for the above approach` `import` `java.lang.*;` `import` `java.util.*;`   `class` `GFG{` `    `  `// Function to check if there exists` `// any pair (P, Q) whose Bitwise XOR` `// is greater than the Bitwise XOR` `// of X and Y` `static` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    ``int` `first = (X ^ Y);` `    ``int` `second = (X + Y);`   `    ``// Check for the invalid condition` `    ``if` `(first == second)` `    ``{` `        ``System.out.println(``"No"``);` `    ``}`   `    ``// Otherwise,` `    ``else` `    ``{` `        ``System.out.println(``"Yes"``);` `    ``}` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{ ` `    ``int` `A = ``2``, B = ``4``;` `    `  `    ``findWinner(A, B);` `}` `}`   `// This code is contributed by offbeat`

## Python3

 `# Python3 program for the above approach`   `# Function to check if there exists` `# any pair (P, Q) whose Bitwise XOR` `# is greater than the Bitwise XOR` `# of X and Y` `def` `findWinner(X, Y):` `    `  `    ``first ``=` `(X ^ Y)` `    ``second ``=` `(X ``+` `Y)`   `    ``# Check for the invalid condition` `    ``if` `(first ``=``=` `second):` `        ``print` `(``"No"``)`   `    ``# Otherwise,` `    ``else``:` `        ``print` `(``"Yes"``)`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    `  `    ``A, B ``=` `2``, ``4` `    `  `    ``findWinner(A, B)`   `# This code is contributed by mohit kumar 29`

## C#

 `// C# program for the above approach` `using` `System;`   `class` `GFG{` `    `  `// Function to check if there exists` `// any pair (P, Q) whose Bitwise XOR` `// is greater than the Bitwise XOR` `// of X and Y` `static` `void` `findWinner(``int` `X, ``int` `Y)` `{` `    ``int` `first = (X ^ Y);` `    ``int` `second = (X + Y);`   `    ``// Check for the invalid condition` `    ``if` `(first == second)` `    ``{` `        ``Console.Write(``"No"``);` `    ``}`   `    ``// Otherwise,` `    ``else` `    ``{` `        ``Console.Write(``"Yes"``);` `    ``}` `}`   `// Driver code` `public` `static` `void` `Main(String[] args)` `{ ` `    ``int` `A = 2, B = 4;` `    `  `    ``findWinner(A, B);` `}` `}`   `// This code is contributed by shivanisinghss2110`

## Javascript

 ``

Output:

`No`

Time Complexity: O(1)
Auxiliary Space: O(1)

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