# Simplifying the Boolean expression $\overline{(a+b)\cdot \overline{a(b\oplus c)}}$ to have lowest number of logical gates

I have this boolean expression $$f(a,b,c)_{Question}=nand(or(a,b) , nand(a, xor(b,c))) = \overline{(a+b)\cdot \overline{a(b\oplus c)}}$$ when I try to simplify the gates using boolean algebra, I get $$f(a,b,c)_{Question}=or(nor(a,b) , and(a,xor(b,c)))=\overline{a+b}+(a\cdot(b⊕c))$$

I need to simplify further and get $$f(a,b,c)_{answer}=a\oplus(b \oplus \overline{a\cdot c})$$ my function: $$f_{answer}$$ is justified by the criteria:

• $$f_{answer}$$ uses lowest number of logical gates,(in our case minimum:3(xor,xor,and))
• there may be n number of possible $$f_{answer}$$, example we can have another answer with same-minimum number of logical gates: $$(a\oplus b) \oplus \overline{a\cdot c}$$ is another possible answer,to our function,and it will be accepted,since it has only 3 logical gates,its is accepted

Question: So how can I find $$f_{answer}$$ when $$f_{Question}$$ is provided, what are the steps

but don't know what's the steps*,can anyone help?

I found these rules for xor: from here but they are of no use

### Edit:

when I expand** $$f(a,b,c)_{Question}=\overline{a+b}+a\cdot(b⊕c)$$
I get $$f(a,b,c)_{expanded}=\overline{(\overline{\overline{a}\overline{b}})\cdot (\overline{a\cdot\overline{\overline{b}\overline{c}}\cdot \overline{bc}})}$$

*The process to simplify,like "first remove brackets...", "Steps" is the answer I need for this question

**Which I think would be the first step in converting any, $$f_{Question}$$ to $$f_{answer}$$

• What does "a xor (b,c)" mean, which appears just after your f(a,b,c) as part of the definition of f? [I would think xor is a binary connective, so should only appear between two boolean expressions, but I don't think (b,c) counts as a boolean expression,] May 28, 2022 at 5:25
• sorry for confusion,i mean xor(b,c) as b xor c and its not "a xor (b,c)",its "nand(a, xor (b,c))" which means !(a*(b xor c)), i think words may confuse,but you can understand the equation on right which uses symbols I guess May 28, 2022 at 5:37