13.3 Venn Diagrams

To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18^th century. These illustrations now called Venn Diagrams.

Note

Venn Diagram

A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.

import numpy as np
import matplotlib.pyplot as plt

# EDIT these two lines. a is True inside circle A, b is True inside circle B.
NAME = "A union B"
region = lambda a, b: a or b
# others to try:
#   "A inter B"        lambda a, b: a and b
#   "Ac inter B"       lambda a, b: (not a) and b
#   "(A union B)c"     lambda a, b: not (a or b)
#   "only one of them" lambda a, b: a != b

n = 500
x, y = np.meshgrid(np.linspace(-2.6, 2.6, n), np.linspace(-1.8, 1.8, n))
inA = (x + 0.6) ** 2 + y ** 2 <= 1.2 ** 2
inB = (x - 0.6) ** 2 + y ** 2 <= 1.2 ** 2
shaded = np.vectorize(region)(inA, inB)

fig, ax = plt.subplots(figsize=(6.4, 4.4))
ax.imshow(shaded, extent=(-2.6, 2.6, -1.8, 1.8), origin="lower",
          cmap="Blues", vmin=0, vmax=1.6, alpha=0.85, aspect="equal")
for cx, lab in ((-0.6, "A"), (0.6, "B")):
    ax.add_patch(plt.Circle((cx, 0), 1.2, fill=False, lw=2, color="black"))
    ax.text(cx * 1.9, 1.25, lab, fontsize=15, ha="center")
ax.add_patch(plt.Rectangle((-2.55, -1.75), 5.1, 3.5, fill=False, lw=1.5, color="black"))
ax.text(-2.4, 1.5, "U", fontsize=13)
ax.set_title(f"shaded region: {NAME}", fontsize=13)
ax.set_xticks([]); ax.set_yticks([])
plt.show()

# The four regions of a 2-set diagram, and whether your expression keeps each one:
labels = {(True, True): "in both (the overlap)", (True, False): "in A only",
          (False, True): "in B only", (False, False): "in neither (outside both circles)"}
for key, text in labels.items():
    print(f"  {text:<34} {'SHADED' if region(*key) else 'not shaded'}")

Basic Venn diagrams can illustrate the interaction of two or three sets.

import numpy as np
import matplotlib.pyplot as plt

# EDIT: which of the 8 regions belongs to your expression?
NAME = "(H inter F) inter Wc"        # the book's outlined region
region = lambda h, f, w: h and f and not w
# others to try:
#   "H union F union W"   lambda h, f, w: h or f or w
#   "A union (B inter Cc)" style: lambda h, f, w: h or (f and not w)
#   exactly one set:      lambda h, f, w: (h + f + w) == 1

n = 500
x, y = np.meshgrid(np.linspace(-2.4, 2.4, n), np.linspace(-2.1, 2.4, n))
centres = [(-0.62, 0.42, "H"), (0.62, 0.42, "F"), (0.0, -0.62, "W")]
masks = [((x - cx) ** 2 + (y - cy) ** 2) <= 1.15 ** 2 for cx, cy, _ in centres]
shaded = np.vectorize(region)(*masks)

fig, ax = plt.subplots(figsize=(5.6, 5.4))
ax.imshow(shaded, extent=(-2.4, 2.4, -2.1, 2.4), origin="lower",
          cmap="Greens", vmin=0, vmax=1.6, alpha=0.85, aspect="equal")
for cx, cy, lab in centres:
    ax.add_patch(plt.Circle((cx, cy), 1.15, fill=False, lw=2, color="black"))
    ax.text(cx * 1.75, cy * 1.75 + 0.25, lab, fontsize=15, ha="center")
ax.set_title(f"shaded region: {NAME}", fontsize=13)
ax.set_xticks([]); ax.set_yticks([])
plt.show()

print("the 8 regions of a three-circle diagram")
print(f"{'H':>3}{'F':>3}{'W':>3}   region                          shaded?")
for h in (True, False):
    for f in (True, False):
        for w in (True, False):
            names = [n for n, v in (("H", h), ("F", f), ("W", w)) if v]
            desc = ("in all three" if len(names) == 3 else
                    " and ".join(names) + " only" if names else "outside all three")
            mark = "SHADED" if region(h, f, w) else "-"
            print(f"{str(h)[0]:>3}{str(f)[0]:>3}{str(w)[0]:>3}   {desc:<30}  {mark}")

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.