##### 4. [2 + 1 + 1 = 4 marks] Given the following expression:

`𝐸 = 𝑥𝑦′(𝑥+𝑦)`

Notes:
`Lec3 slide 24 sum/products`
`Lec3 slide 28 simplify`

`In the sum-of-products form, ANDed variables are ORed together. eg.`
`F(x,y,z) = xy + xz + yz`
`In the product-of-sums form, ORed variables are ANDed together. eg `
`F(x,y,z) = (x+y)(x+z)(y+z)`

- a) ⭐ Draw the circuit that represents this expression as is.
![](q4a.png)
`E = xy'(x+y)`

b) ⭐ The expression can be simplified to sum of products using one rule: which rule is it?

a = x
b = ~y
c = (x+y)

(a * b) * c = a * (b * c) (associative law)

(x * ~y) * (x + y) (associative law)
= x * (~y * (x + y)) (involution law)
= x * (x~y) (idempotent law)
= x * ~y

c) ⭐ Draw the circuit that represents the simplified expression.
![](q4c.png)