a$)$ Prove the two properties mentioned below:

i$.$ Absorption Property $[2(1)/(2)$ marks$]$

A$+$A$.$B$=$A

ii$.$ Adjacency Property $[2(1)/(2)$ marks$]$

A$.$B$+$A$.$B$=$A

b$)$ Given a four input system Y$($X$4,$X$3,$X$2,$X$1)$ in the truth table below. Realize the system circuit

diagrams using:

X$4$ X$3$ X$2$ X$1$ Y

$00001$

$00011$

$00100$

$00111$

$01000$

$01010$

$01101$

$01110$

$10001$

$10011$

$10101$

$10111$

$11000$

$11011$

$11101$

$11110$

i$.$ a $8\backslash$times $1$ MUX when the truth tables is flipped about input X$4.$

$[4$ marks$]$

ii$.$ a $4\backslash$times $16$ decoder with low logic output.

$[4$ marks$]$