In the bottom of a large bowl, combine the mayonnaise, vinegar, sugar, celery seeds, pepper, and salt. Whisk the dressing until it is smooth. Set this aside.
Wash and slice the apples into thin slices, then matchsticks, to yield about 2 cups of apple. Add this to the bowl with the dressing.
Slice the fennel bulb in half from tip to root and then into very thin slices to yield about 1½ cups. Add these to the bowl with the apples and dressing.
Cut a head of cabbage in half and then into quarters. Very thinly slice the quarters into shreds to yield about 2 cups of cabbage. Add this to the large bowl.
Shred a carrot using the large holes of a box grater to yield about 1 cup. Add this to the bowl.
Very thinly slice the red onion to yield about ¼ cup. Add this to the bowl.
Toss all of the vegetables with the dressing until evenly coated. Transfer to a serving dish.
Nutrition facts are sometimes provided below and are calculated using an online calculator. With specific brands of ingredients and additions, omissions, or substitutions the nutrition facts may change. We encourage you to use your own nutrition facts calculator to obtain the most accurate nutrition facts for your meal.