Wash, peel and cut potatoes into bite-size pieces. In a large pot, bring water to a boil. Add potatoes and cook until the potatoes are tender. Drain the potatoes and put them in a big bowl.

Put the eggs in another saucepan and add water to cover them. Bring the water to a boil, and then let the water and eggs boil gently for 5 minutes. Drain the eggs and cover them with ice cold water. Allow the eggs to sit in the cold water while you mix up the dressing.

For the dressing, combine the salt, mustard, mayonnaise, vinegar and pepper. Mix until completely combined. Set aside.

Once the eggs are cool enough to touch, drain the water. Peel the eggs and dice them, then add them to the potatoes. Pour the dressing over top the eggs and potatoes and mix until the potatoes are covered.

After the salad is completely cooled, add the diced cucumber. If you add this while the potatoes are still warm, it will soften the cucumber. Add in the onions, celery and carrots, if desired.

Refrigerate until ready to serve.