CH1 - DeMorgan's In Detail

Just as a refresher:

$$\begin{array}{r}(A\cup B{)}^C=A^C\cap B^C\\ (A\cap B{)}^C=A^C\cup B^C\end{array}$$

In general, for a finite/infinite collection of events $A_1,A_2,\dots ,A_n$ :

$$(U_j{A}_j{)}^C=N_j(A_j^C)$$

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Commutativity, Associativity, and Distributive

Commutativity:

The law of commutativity states that the order of the sets in a union or intersection does not matter.

$$A\cup B=B\cup A$$ $$A\cap B=B\cap A$$

Associativity:

The law of associativity states that the grouping of sets in a union or intersection does not matter.

$$(A\cup B)\cup C=A\cup (B\cup C)$$ $$(A\cap B)\cap C=A\cap (B\cap C)$$

Distributive:

The law of distributive states that the union and intersection operations distribute over each other.

$$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$ $$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$$

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Example

Example

A power cell consists of two subcells, each of which can provide from 0 to 5 volts, regardless f what the other subcell provides, The power cell ins functional if and only if the sum of the two voltages of the subcells is at least 6 volts. An experiment consists of measuring and recording the voltages of the 2 subcells.

_Let X be the voltage of the first subcell

Let Y be the voltage of the second subcell._

What is the sample space?

We can find the sample space by listing all possible outcomes of the experiment. Since each subcell can provide from 0 to 5 volts, the sample space is:

$$S=\{(x,y)|0\le x\le 5AND0\le y\le 5\}$$

Let A be the event that the power cell is functional. Define A.

We can find a by listing all the outcomes in which the power cell is functional. The power cell is functional if and only if the sum of the two voltages is at least 6 volts. Therefore, the event A is:

$$A=\{(x,y)\in S|x+y\ge 6\}$$

Let B be the event that two subcells have the same voltage. Define B.

We can find B by listing all the outcomes in which the two subcells have the same voltage. Therefore, the event B is:

$$B=\{(x,y)\in S|x=y\}$$

Let C be the event that the first subcell has a strict higher voltage than the second. Define C

We can find C by listing all the outcomes in which the first subcell has a higher voltage than the second. Therefore, the event C is:

$$C=\{(x,y)\in S:x>y\}$$

Let D be the event that the power cell is not functional but needs less than one additional volt to become functional.

We can find D by listing all the outcomes in which the power cell is not functional but needs less than one additional volt to become functional. Therefore, the event D is:

$$D=\{(x,y)\in S:5<x+y<6\}$$