FUZZY_AND

Computes the fuzzy AND operator, representing the intersection of two fully defined fuzzy sets. As a soft minimum, it returns the point-wise minimum of membership values from both sets overlapping the same universe.

Excel Usage

=FUZZY_AND(x, mfx, y, mfy)

x (list[list], required): Universe variable for the first fuzzy set mfx.
mfx (list[list], required): Membership function values corresponding to x.
y (list[list], required): Universe variable for the second fuzzy set mfy.
mfy (list[list], required): Membership function values corresponding to y.

Returns (dict): Dictionary containing the combined universe points and the intersection membership values array.

Example 1: Fuzzy AND on matching universes

Inputs:

x			mfx			y			mfy
1	2	3	0.5	0.8	1	1	2	3	0.2	0.9	0.5

Excel formula:

=FUZZY_AND({1,2,3}, {0.5,0.8,1}, {1,2,3}, {0.2,0.9,0.5})

Expected output:

{"type":"Double","basicValue":0.2,"properties":{"z":{"type":"Array","elements":[[{"type":"Double","basicValue":1}],[{"type":"Double","basicValue":2}],[{"type":"Double","basicValue":3}]]},"mfz":{"type":"Array","elements":[[{"type":"Double","basicValue":0.2}],[{"type":"Double","basicValue":0.8}],[{"type":"Double","basicValue":0.5}]]}}}

Example 2: Fuzzy AND with overlapping shifted universes

Inputs:

x			mfx			y			mfy
0	1	2	0.1	0.7	1	1	2	3	0.4	0.9	0.2

Excel formula:

=FUZZY_AND({0,1,2}, {0.1,0.7,1}, {1,2,3}, {0.4,0.9,0.2})

Expected output:

{"type":"Double","basicValue":0.1,"properties":{"z":{"type":"Array","elements":[[{"type":"Double","basicValue":0}],[{"type":"Double","basicValue":1}],[{"type":"Double","basicValue":2}],[{"type":"Double","basicValue":3}]]},"mfz":{"type":"Array","elements":[[{"type":"Double","basicValue":0.1}],[{"type":"Double","basicValue":0.4}],[{"type":"Double","basicValue":0.9}],[{"type":"Double","basicValue":0.2}]]}}}

Example 3: Fuzzy AND for singleton fuzzy sets

Inputs:

x	mfx	y	mfy
1	0.4	1	0.9

Excel formula:

=FUZZY_AND(1, 0.4, 1, 0.9)

Expected output:

{"type":"Double","basicValue":0.4,"properties":{"z":{"type":"Array","elements":[[{"type":"Double","basicValue":1}]]},"mfz":{"type":"Array","elements":[[{"type":"Double","basicValue":0.4}]]}}}

Example 4: Fuzzy AND of sparse membership profiles

Inputs:

x			mfx			y			mfy
0	2	4	0	0.6	0.2	0	2	4	0.3	0.4	0.9

Excel formula:

=FUZZY_AND({0,2,4}, {0,0.6,0.2}, {0,2,4}, {0.3,0.4,0.9})

Expected output:

{"type":"Double","basicValue":0,"properties":{"z":{"type":"Array","elements":[[{"type":"Double","basicValue":0}],[{"type":"Double","basicValue":2}],[{"type":"Double","basicValue":4}]]},"mfz":{"type":"Array","elements":[[{"type":"Double","basicValue":0}],[{"type":"Double","basicValue":0.4}],[{"type":"Double","basicValue":0.2}]]}}}

Python Code

Show Code

import numpy as np
from skfuzzy import fuzzy_and as fuzz_and

def fuzzy_and(x, mfx, y, mfy):
    """
    Calculate the fuzzy AND operator (intersection) of two fuzzy sets.

    See: https://pythonhosted.org/scikit-fuzzy/api/skfuzzy.html#skfuzzy.fuzzy_and

    This example function is provided as-is without any representation of accuracy.

    Args:
        x (list[list]): Universe variable for the first fuzzy set `mfx`.
        mfx (list[list]): Membership function values corresponding to `x`.
        y (list[list]): Universe variable for the second fuzzy set `mfy`.
        mfy (list[list]): Membership function values corresponding to `y`.

    Returns:
        dict: Dictionary containing the combined universe points and the intersection membership values array.
    """
    try:
        def to1d(arr):
            if isinstance(arr, list):
                flat = []
                for row in arr:
                    row_list = row if isinstance(row, list) else [row]
                    for val in row_list:
                        try:
                            flat.append(float(val))
                        except (TypeError, ValueError):
                            continue
                return np.array(flat)
            return np.array([float(arr)])

        x_arr = to1d(x)
        mfx_arr = to1d(mfx)
        y_arr = to1d(y)
        mfy_arr = to1d(mfy)

        if len(x_arr) == 0 or len(mfx_arr) == 0 or len(y_arr) == 0 or len(mfy_arr) == 0:
            return "Error: Input arrays cannot be empty"

        if len(x_arr) != len(mfx_arr):
            return "Error: x and mfx must have the same number of elements"

        if len(y_arr) != len(mfy_arr):
            return "Error: y and mfy must have the same number of elements"

        z_arr, mfz_arr = fuzz_and(x_arr, mfx_arr, y_arr, mfy_arr)

        return {
            "type": "Double",
            "basicValue": float(mfz_arr[0]) if len(mfz_arr) > 0 else 0.0,
            "properties": {
                "z": {
                    "type": "Array",
                    "elements": [[{"type": "Double", "basicValue": float(val)}] for val in z_arr]
                },
                "mfz": {
                    "type": "Array",
                    "elements": [[{"type": "Double", "basicValue": float(val)}] for val in mfz_arr]
                }
            }
        }
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

x *

Universe variable for the first fuzzy set `mfx`.

mfx *

Membership function values corresponding to `x`.

y *

Universe variable for the second fuzzy set `mfy`.

mfy *

Membership function values corresponding to `y`.