Use Boolean laws to simplify the following expressions:

i$.\frac{?}{\text{b}\text{a}\text{r}}(((\frac{?}{\text{b}\text{a}\text{r}}(A))*(\frac{?}{\text{b}\text{a}\text{r}}(B)))\frac{+}{b}ar((A+C)))$

ii$.\frac{?}{\text{b}\text{a}\text{r}}((\frac{?}{\text{b}\text{a}\text{r}}(A+B))\frac{+}{b}ar((A*(A+B))))$

A device accepts four inputs $P,Q,R,S$ which represents the natural binary numbers in the range $0000$ to $1111$ that represent a decimal number $0$ to $15.$ P represents the most significant bit position. The output, $F$ of the circuit is true if the input to the circuit represents a prime number and is false otherwise.

i$.$ Construct a truth table to show the behaviour for this device and hence determine the SOP expression for this device.

ii$.$ Use the Karnaugh map method to simplify the expression you got in i$.$

Minimize the function $f(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}m(3,7,10,11,13,15)$ using the Quine$-$McCluskey method.

A traditional children's riddle concerns a farmer traveling with a sack of rye, a goose, a mischievous dog. The farmer comes to a river that he must cross from east to west. A boat is available, but only has room for the farmer and one of his possessions. If the farmer is not present, the goose will eat the rye or the dog will eat the goose. We wish to design a circuit to simulate the conditions of this riddle. A separate switch is provided for the farmer, the rye, the goose and the dog. Each switch has two states, depending on whether the corresponding object is on the east bank or west bank of the river. The rules of play stipulate that no more than two switches be moved at the same time and that the farmer must move $($ to row the boat$)$ each time switches are moved. The switch for the farmer provides a signal $F,$ which is high if the farmer is on the east bank and low if he is on the west bank. Similar logic signals $(G$ for the goose, $D$ for the dog and $R$ for the rye$)$ are high corresponding object is in the east bank and low if it is on the west bank. Construct the truth table for this riddle $($consider the output signal to be $Z$ the output is $1$ if there is danger and $0$ if the is not danger$)$ and draw the $K$ map of its POS expression.

With an aid of a diagram explain operations of a $4$ to $1-$ line multiplexer,