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# Summary of fuzzy set theory article

The article discusses how fuzzy set theory concepts can be used in medical diagnosis. It expands on the notions of fuzzy sets and describes how they might be used to improve artificial intelligence in medicine.

The Use of Fuzzy Set Theory in Medicine

Fuzzy systems based on fuzzy set theory can be used as function approximators in the development of a model-based controller for closed loop muscle relaxant delivery.

Question 2: What are the logical connectives for the aforementioned propositions?

Let p and q represent propositions.

It is warm.

q: The sky is blue.

It follows that the following applies:

It is warm

q: It is sunny

It follows that the following applies:

It is warm and sunny

p q

It is warm but not sunny

p q

It is not warm and it is not sunny

p q

It is either sunny or warm (or both) - inclusive

p q

If it is warm, it is also sunny

p q

Either it is warm or it is sunny but it is not sunny if it is warm

(p q) (p q)

That it is warm is necessary and sufficient for it to be sunny

p q

Question 3

Truth table

p

q

p∨q

p∧q

¬(p∧q)

(p∨q)∧¬(p∧q)

T

T

T

F

F

F

T

F

T

F

T

T

F

T

T

F

T

T

F

F

F

T

T

F

Question 4

Application of the PIE and Subtraction rule

The subtraction rule:

If a task can be done in either n1 or n2 ways, then the number of ways to do the task is nothing but the difference between n1+ n2 and the number of ways to do the task that are common to both n1 and n2.

Example: It is warm but not sunny

Let

n1: the number of warm days

n2: the number of days that it is sunny

It follows that

| n1 n2 | = | n1 | + | n2 | - | n1 n2 |

The number of warm but not sunny days is equal to the difference between the sum of the number of warm and sunny days and the number of days that it is both warm and sunny.

Question 5

Relation of the principle of inclusion and exclusion to rules of manipulation and simplification of logic and to combinatorics

The principle of inclusion and exclusion is related to the rules of manipulation and simplification of logic in that,

For two events a1 and a2 with possible outcomes p1 and p2 respectively,

If both events are mutually exclusive (can’t occur at the same time), the possible outcomes of both events occur is given by the sum rule: p1 + p2

If both events are not mutually exclusive (can occur at the same time), the possible outcomes of both events occur is given by the product rule: p1. p2

However, if only one of the events can occur, then neither the product nor the sum rule applies. Instead, we use the principle of inclusion and exclusion. Based on this principle, it follows that

we have to count the number of possible outcomes of p1 and p2 minus the number of possible outcomes common in both (Rosen, 2012).

As such, considering the possible outcomes as sets, the following holds:

| p1 p2 | = | p1 | + | p2 | - | p1 p2 |

Question 6

Proof of Idempotent laws

For the following Boolean algebra example,

The following laws identify each of the steps

Step 1 = a + 0 = identity law

Step 2 = further simplification of the identity using its complement

Step 3 = distributive law

Step 4 = identity law

Step 5 = the proof that a = a + a

Question 7

Proof of Idempotent laws

For a given element x, xx = x and xx = x hold for every x.

Proof:

For the idempotent law xx = x, it follows that,

xx = x

Let x = {x | xx}

xx = {x | xx xx}

= {xx}

= x

xx = x

Let x = {x | xx}

xx = {x | xx xx}

= {xx}

= x

References:

Rosen, K. (2012). Discrete mathematics and its applications (7th ed.). New York: McGraw-Hill.