Methane biofilters can be used to oxidize methane using biological activities. It has become necessary to compare performance of two test columns, A and B$.$ The methane outflow level at the surface, in nondimensional units of X $=\{50, 100, 150, 200],$ was detected and is tabulated below against the respective methane inflow into each test column. The following fuzzy sets represent the test columns:

Calculate the union, intersection, and the difference for the test columns.

Union of A and BIntersection of A and BDifferent between A and B

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$$QUESTION $3,$ MARKS: $12$

We want to compare two sensors based upon their detection levels and gain settings. For a universe of discourse of gain settings, X $=\{0, 20, 40, 60, 80, 100\},$ the sensor detection levels for the monitoring of a standard item provides typical membership functions to represent the detection levels for each of the sensors; these are given below in standard discrete form:

Find the following membership functions using standard fuzzy operations:

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$$QUESTION $4,$ MARKS: $10$

In$$mechanics$,$the energy$$of a$$moving$$body$$is called$$kinetic energy. If an$$object$$of$$mass$$m$($kilograms$)$is$$moving$$with a$$velocity$$v$($meters$$per$$second$),$ Suppose$$we$$model$$the$$mass and$$velocity as$$inputs$$to a$$system$($moving$$body$)$ and$$the$$energy as$$output$,$then$$observe$$the$$system for a$$while and$$deduce$$the following$$two$$rules$$of$$inference$$based$$on$$our$$observations:

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Rule $1$:$$IF$$x$1$is$$small $$and$$x$2$is$$high$$velocity $,$ THEN$$y$$is$$medium energy

Rule$2$:$$IF$$x$1$is$$large$$mass$$or$$x$2$is$$medium velocity,$$THEN$$y$$is$$high energy.

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Let$$input$($i$) = 0\mathrm{.}35$kg $($mass$)$ and$$input$($j$) = 55$m$/$s $($velocity$),$ Produce$$the$$output$$using a Mamdani implication $($use$$Centroid$,$and$$Weighted average$$method for$$defuzzification$)$