Simplifying a term

**shadow147** · October 23rd 2009, 02:23 AM

Hello,

Can anyone help me simplifying this term?

a AND (a xor b)

I know that I can write it like this:

(a AND not b) OR (not a AND b)

But how to simplify further? Would be grateful for any help. Thanks!

**PiperAlpha167** · October 23rd 2009, 06:21 AM

Originally Posted by shadow147

Hello,

Can anyone help me simplifying this term?

a AND (a xor b)

I know that I can write it like this:

(a AND not b) OR (not a AND b)

But how to simplify further? Would be grateful for any help. Thanks!

(a AND not b) OR (not a AND b)

This is just equivalent to (a xor b), not the initial formula.
But, it turns out that all you need is the left disjunct.
That's the (perfect) disjunctive normal form for the initial formula.

**Soroban** · October 23rd 2009, 12:47 PM

Hello, shadow147!

I'll let you supply some of the reasons . . .

Simplify: .a AND (a XOR b)

. . $\begin{array}{ccc}a \cap \bigg[(a \cap b') \cup (a' \cap b)\bigg] & & \text{Given} \\ \\ \bigg[ a\cap (a \cap b')\bigg] \cup \bigg[a \cap(a'\cap b)\bigg] && \text{Distributive} \\ \\ \bigg[(a\cap a) \cap b'\bigg] \cup \bigg[(a\cap a') \cap b\bigg] & & \text{Associative}\end{array}$

. . . . . $\begin{array}{c}(a\cap b') \cup (f \cap b) \\ \\(a \cap b') \cup f \\ \\ a \cap b' \end{array}$

**shadow147** · October 23rd 2009, 02:10 PM

Awesome, much appreciated, thank you!
Just one question:

$ a\cap a' $

Thats = 0 isn't it? Why did you replace it with f? And at the end you got

$ a\cap b' \cup f $

Why is it possible to ignore the f and say that

$ (a\cap b') $

is the result?

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