The Karnaugh map, also known as the K-map, is a method to simplify boolean algebra expressions.

kmap.cs

using System;

class MainClass
{
	// The Karnaugh map, also known as the K-map, is a method to simplify boolean algebra expressions.
	public static void Main(string[] args)
	{
		// kmap cells
		var kmap = new bool[4];

		// kmap variables
		string[] variables =
		{
			"x",
			"y"
		};

		Console.Write("Input must contain either TrueString or FalseString; otherwise, false\n");
		Console.Write("The comparison is case-insensitive\n");
		Console.Write("TrueString: \"True\"\n");
		Console.Write("FalseString: \"False\"\n");
		Console.Write("\n");

		//		 Y  !Y
		//	   ---------
		//	X  | 0 | 1 |
		//	   ---------
		// !X  | 2 | 3 |
		//	   ---------

		// user input
		for (var i = 0; i < kmap.Length; i++)
		{
			Console.Write("Input for cell {0}: ", i);

			// must contain either TrueString or FalseString; otherwise, an exception is thrown.
			// The comparison is case-insensitive.
			// all other strings will be represented as false
			Boolean.TryParse(Console.ReadLine(), out kmap[i]);
		}
		Console.Write("\n");

		// Print the cell listing
		for (var i = 0; i < kmap.Length; i++)
		{
			Console.WriteLine("Cell {0}: {1}", i, kmap[i]);
		}
		Console.Write("\n");

		// If everything is false the function must be false / empty / 0
		if (!kmap[0] && !kmap[1] && !kmap[2] && !kmap[3])
		{
			Console.WriteLine("This is an empty function");
			return;
		}

		// If everything is true the function must be true / 1
		if (kmap[0] && kmap[1] && kmap[2] && kmap[3])
		{
			Console.WriteLine("the value of this function is 1");
			return;
		}

		// just some polish
		Console.Write("F(x, y) = ");

		// Y column
		if (kmap[0])
		{
			Console.Write("{0}{1}", variables[0], variables[1]);

			// check our neighbors
			if (kmap[1] || kmap[2])
			{
				Console.Write("+");
			}
		}

		// !Y Column
		if (kmap[1])
		{
			Console.Write("{0}!{1}", variables[0], variables[1]);
			if (kmap[0] || kmap[3])
			{
				Console.Write("+");
			}
		}

		// X row
		if (kmap[2])
		{
			Console.Write("!{0}{1}", variables[0], variables[1]);
			if (kmap[3])
			{
				Console.Write("+");
			}
		}

		// !X row
		if (kmap[3])
		{
			Console.Write("!{0}!{1}", variables[0], variables[1]);
			if (kmap[0] || kmap[3])
			{
				// we dont care about our neihbors
				// this is the last cell and nothing will
				// come after
			}
		}
	}
}