$2$ Suppose we have eight machines that we want to monitor with respect to their availability. We can keep track of them with an eight bit BUSYNESS bit vector, where a bit is $1$ if the unit is free and $0$ if the unit is busy. The bits are labeled, from right to left, from $0$ to $7.$ The BUSYNESS bit vector I$1000010$ corresponds to the situation where only units $7,6,$ and $1$ are free and therefore available for work assignment.

Suppose work is assigned to unit $7.$ We update our BUSYNESS bit vector by performing the logical AND, where our two sources are the current bit vector $11000010$ and the bit mask $01111111.$ The purpose of the bit mask is to clear bit $7$ of the BUSYNESS bit vector, while leaving alone the values corresponding to all the other units. The result is the bit vector $01000010,$ indicating that unit $7$ is now busy.

Suppose unit $5$ finishes its task and becomes idle. We can update the BUSYNESS bit vector by performing the logical OR of it with the bit mask $00100000.$ The result is $01100010,$ indicating that unit $5$ is now available.

Now based on the above data, answer the following questions.

$(6$ Points$)$

a$.$ What mask value and what operation would one use to indicate that machine $2$ is busy?

b$.$ What mask value and what operation would one use to indicate that machines $2$ and $6$ are longer busy? $($Note: this can be done with only one operation.$)$

c$.$ What mask value and what operation would one use to indicate that all machines are busy?

d$.$ What mask value and what operation would one use to indicate that all machines are idle?

e$.$ Develop a procedure to isolate the status bit of machine $2$ as the sign bit. For example, if the BUSYNESS pattern is $01011100,$ then the output of this procedure is $10000000.$ If the BUSYNESS pattern is $01110011,$ then the output is $00000000.$ In general, if the BUSYNESS patrern is b$7$ b$6$ b $5$ b $4$ b $3$ b$2$ b$1$ b$0,$ then the output is b$2000000.$ Hint: What happens when you ADD a bit pattern to itself?