C++ - Implement a string class with basic functionality

// ------String.h------

#ifndef STRING_H
#define STRING_H

#include <iostream>

using namespace std;

class String
{
public:
    String();
    String(int n, char c);
    String(const char* source);
    String(const String& s);
    // String& operator = (char* s);
    String& operator = (const String& s);
    ~String();

    char& operator [] (int i){return a[i];}
    const char& operator [] (int i) const {return a[i];} // Indexer to constant.
    String& operator += (const String& s);
    int length();

    friend istream& operator >> (istream& is, String& s); // Try to figure out why set ">>" friend function member.
    // friend bool operator < (const String& left,const String& right);
    friend bool operator > (const String& left, const String& right); // These three operators is not necessary to set them friend. Here is just for simplification.
    friend bool operator == (const String& left, const String& right);
    friend bool operator != (const String& left, const String& right);
private:
    char* a;
    int size;
};

#endif

// ------String.cpp------

#include "String.h"
#include <cstring>
#include <cstdlib>

String::String()
{
    a = new char[1];
    a[0] = '\0';
    size = 0;
}

String::String(int n, char c)
{
    a = new char[n + 1];
    memset(a, c, n);
    a[n] = '\0';
    size = n;
}

String::String(const char* source)
{
    if(source == NULL)
    {
        a = new char[1];
        a[0] = '\0';
        size = 0;
    }
    else
    {   
        size = strlen(source);
        a = new char[size + 1];
        strcpy(a, source);
    }
}

String::String(const String& s)
{
    size = strlen(s.a);
    a = new char[size + 1];
    strcpy(a,s.a);
}

String& String::operator = (const String& s)
{
    if(this == &s)
        return *this;
    else
    {
        delete[] a;
        size = strlen(s.a);
        a = new char[size + 1];
        strcpy(a, s.a);
        return *this;
    }
}

String::~String()
{
    delete[] a;
}

String& String::operator += (const String& s)
{
    int j = strlen(a);
    int size = j + strlen(s.a);
    char* tmp = new char[size + 1];
    strcpy(tmp, a);
    strcpy(tmp + j, s.a);
    delete[] a;
    a = tmp;

    return *this;
}

int String::length()
{
    return strlen(a);
}

// ------Main.cpp------

#include <iostream>
#include "String.h"

using namespace std;

bool operator == (const String& left, const String& right)
{
	int a = strcmp(left.a, right.a);
	if(a == 0)
	{
		return true;
	}
	else
	{
		return false;
	}
}

bool operator != (const String& left, const String& right)
{
	return !(left == right);
}

ostream& operator << (ostream& os, String& s)
{
	int length = s.length();
	for(int i = 0; i < length; i++)
	{
		os << s[i];
	}

	return os;
}

String operator + (const String& a, const String& b)
{
	String temp;
	temp = a;
	temp += b;
	return temp;
}

bool operator < (const String& left, const String& right)
{
	int j = 0;
	while((left[j] != '\0') && (right[j] != '\0'))
	{
		if(left[j] < right[j])
		{
			return true;
		}
		else
		{
			if(left[j] == right[j])
			{
				j++;
				continue;
			}
			else
			{
				return false;
			}
		}
	}

	if((left[j] == '\0') && (right[j] != '\0'))
	{
		return true;
	}
	else
	{
		return false;
	}
}

bool operator > (const String& left, const String& right)
{   
	int a = strcmp(left.a, right.a);
	if(a > 0)
	{
		return true;
	}
	else
	{
		return false;
	}
}

istream& operator >> (istream& is, String& s)
{
	delete[] s.a;
	s.a = new char[20];
	int m = 20;
	char c;
	int i = 0;
	while(is.get(c) && isspace(c));
	if(is)
	{
		do
		{
			s.a[i] = c;
			i++;
			if(i == m - 1)
			{
				s.a[i] = '\0';
				char* b = new char[m];
				strcpy(b, s.a);
				m = m * 2;
				s.a = new char[m];
				strcpy(s.a, b);
				delete[] b;
			}
		}
		while (is.get(c) && !isspace(c));

		if(is)
		{
			is.unget();
		}
	}

	s.size = i;
	s.a[i] = '\0';
	return is;
}

int main()
{
	String a = "abcd";
	String b = "www";
	String c(6, 'l');
	String d;
	String e = a;
	String f;
	cin >> f;
	String g;
	g = a + b;

	if(a < b)
	{
		cout << "a < b" << endl;
	}
	else
	{
		cout << "a >= b" << endl;
	}

	if(e == a)
	{
		cout << "e == a" << endl;
	}
	else
	{
		cout << "e != a" << endl;
	}

	b += a;
	cout << a << endl;
	cout << b << endl;
	cout << c << endl;
	cout << d << endl;
	cout << e << endl;
	cout << f << endl;
	cout << g << endl;
	cout << g[0] << endl;

	return 0;
}

// Output:
/*
t
a < b
e == a
abcd
wwwabcd
llllll

abcd
t
abcdwww
a
*/