For a triangular membership function with parameters a=1, b=5, and c=9, if b is changed to 7, which of the following statements are true: At x=3 and x=7, the degree of membership will be 0.5 At x=4 and x=8, the degree of membership will be 0.5 At x=5 and x=7, the degree of membership will be 0.5 None of the above

The triangular membership function is defined as follows:

f(x;a,b,c) = 0, if x <= a or x >= c f(x;a,b,c) = (x-a)/(b-a), if a <= x <= b f(x;a,b,c) = (c-x)/(c-b), if b <= x <= c

Given the parameters a=1, b=7, and c=9, we can calculate the degree of membership for the given x values:

At x=3, f(x;1,7,9) = (3-1)/(7-1) = 2/6 = 0.33 At x=7, f(x;1,7,9) = (9-7)/(9-7) = 1

At x=4, f(x;1,7,9) = (4-1)/(7-1) = 3/6 = 0.5 At x=8, f(x;1,7,9) = (9-8)/(9-7) = 1/2 = 0.5

At x=5, f(x;1,7,9) = (5-1)/(7-1) = 4/6 = 0.67 At x=7, f(x;1,7,9) = (9-7)/(9-7) = 1

So, the statement "At x=4 and x=8, the degree of membership will be 0.5" is true.

The triangular membership function is a type of function used in fuzzy logic and fuzzy systems to represent a fuzzy set. It is defined by three parameters: a, b, and c, which represent the "feet" and "peak" of the triangle, respectively.

The degree of membership of an element x in a fuzzy set represented by a triangular membership function is calculated as follows:

1. If x < a or x > c, then the degree of membership is 0.
2. If a <= x < b, then the degree of membership is (x - a) / (b - a).
3. If b <= x <= c, then the degree of membership is (c - x) / (c - b).

Now, let's apply these rules to the given parameters and x values:

1. For a=1, b=7, and c=9, at x=3: Since 1 <= 3 < 7, the degree of membership is (3 - 1) / (7 - 1) = 2 / 6 = 0.33 (approximately), not 0.5.

2. For a=1, b=7, and c=9, at x=7: Since 7 <= 7 <= 9, the degree of membership is (9 - 7) / (9 - 7) = 1, not 0.5.

3. For a=1, b=7, and c=9, at x=4: Since 1 <= 4 < 7, the degree of membership is (4 - 1) / (7 - 1) = 3 / 6 = 0.5. So, this statement is true.

4. For a=1, b=7, and c=9, at x=8: Since 7 <= 8 <= 9, the degree of membership is (9 - 8) / (9 - 7) = 1 / 2 = 0.5. So, this statement is also true.

5. For a=1, b=7, and c=9, at x=5: Since 1 <= 5 < 7, the degree of membership is (5 - 1) / (7 - 1) = 4 / 6 = 0.67 (approximately), not 0.5.

6. For a=1, b=7, and c=9, at x=7: As calculated above, the degree of membership is 1, not 0.5.

So, the correct answer is: "At x=4 and x