Question

In a water treatment process, we use a biological process to remove biodegradable organic matter. The organic matter is measured as the BOD, where the optimal BOD of effluent should be less than 20 mg l⁻¹. Let B represent a fuzzy set good effluent on the universe of optical BOD values (20, 40, 60) as defined by the membership function

μB = (0.5/60) + (0.8/40) + (1.0/20)

The retention time is critical to a bioreactor, we try to find the retention time, measured in days. Let T represent a fuzzy set called optimal retention time on the universe of days (6, 8, 10) as given by the membership function

μₜ = (0.9/10) + (0.6/8) + (0.4/6)

The utilization rate of organic food indicates the level of the treatment in the biological process, and this rate is measured on a universe of fractions from 0 to 1. where 1 is optimal. Fuzzy set U will represent high utilization rates, as defined by the membership function

μᵤ = (1/0.9) + (0.8/0.8) + (0.6/0.7) + (0.4/0.6)

We can define the following relations:
R= B x T. which reflects how retention time affects BOD removal:
S= T x U, which relates how retention time affects organic food consumption;
W= R∘S, which represents the BOD removal and food utilization.

Find R and using Cartesian products.

Answer 1

In a water treatment process, the biological process is used to remove organic matter. The fuzzy sets B and T represent good effluent and optimal retention time, respectively. The relations R, S, and W relate retention time to BOD removal and food utilization.

Step-by-step explanation:

In a water treatment process, the biological process is used to remove biodegradable organic matter, measured as the biochemical oxygen demand (BOD). The optimal BOD of effluent should be less than 20 mg/L. The fuzzy set 'B' represents good effluent with an optimal BOD, and the fuzzy set 'T' represents optimal retention time. The relations 'R', 'S', and 'W' relate retention time to BOD removal and food utilization.

B x T (R) reflects how retention time affects BOD removal.

T x U (S) relates how retention time affects organic food consumption.

R∘S (W) represents the BOD removal and food utilization.