I have no idea how to do boolean algebra,
First question is
x'y + x(x + y')
I need to first draw a circuit diagram(logic gate) and then simplify it and draw a simplified logic gate.
As of now I only have it down to
x'y + xx + xy'
Am I right in simplifying it down to
x'y + xy' = 0 as x+x' = 0 and y+y' = 0 ?? and then having + xx which is equivalent to x?
Ending up with the answer of 0 + 0 + x which = x?
In the comments I already simplified it to
$$\bar{x}y + x + x\bar{y}$$
(distribute $x$ into the parenthesis, then by tautology $xx = x$)
Pull out an $x$ on the right side.
$$\bar{x}y + x + x\bar{y}$$ $$\bar{x}y + x(1 + \bar{y})$$ $$\bar{x}y + x(1)$$ $$\bar{x}y + x$$
We can analyse this expression by the means of the truth table, i.e., we ask ourselves when this expression becomes true. This means: If $x = \text{True}$, then the whole expression becomes true. If $x =\text{False}$, then $\bar{x} = 1$ and the expression only becomes $y$. Then $y = 1$ makes the expression true.
$\Longrightarrow$ If $x = 1$ the expression is true
$\Longrightarrow$ If $y = 1$ the expression is true
$\Longrightarrow$ The expression cannot be made true through other means
$\Longrightarrow$ The whole expression equals $$x + y$$