Boolean Equivalent expressions

[autumn siren]

i need to verify if the two expressions below are equivalent.
𝑥𝑦̅𝑧 + 𝑥𝑦̅𝑧̅ + 𝑥̅𝑦̅𝑧̅ and 𝑥𝑦̅ + 𝑦̅𝑧̅.
i have done it to this point

𝑥.𝑦̅ + 𝑥.𝑦̅.(z + 𝑧̅) + 𝑥̅.𝑦̅.𝑧̅ (distribute over + )
𝑥.𝑦̅ + 𝑥.𝑦̅.1 + 𝑥̅.𝑦̅.𝑧̅ (z + 𝑧̅ = 1 )
𝑥.𝑦̅ + 𝑥.𝑦̅ + 𝑥̅.𝑦̅.𝑧̅ (y.1 = 1.y = y )
𝑥.𝑦̅ + 𝑦̅ + (𝑥+𝑥̅).𝑦̅.𝑧̅ ( distribute over + )
𝑥.𝑦̅ + 𝑦̅ + 1.𝑦̅.𝑧̅ (x + 𝑥̅ = 1 )
𝑥.𝑦̅ + 𝑦̅ + 𝑦̅.𝑧̅ (x.1 = 1.x = x )

and i know the answer is 𝑥.𝑦̅ + 𝑦̅.𝑧̅
but im not sure what the actual rule is used to remove the extra 𝑦̅ and i need to give it for each step
am i missing something simple?

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[mild tide]

are you sure the formula is correct? like for example the 1st line

[autumn siren]

yes i think it is

and i have a table showing it is 𝑥.𝑦̅ + 𝑦̅.𝑧̅

[mild tide]

may Ik what formula you used for the 1st line

[autumn siren]

there is no specific formula i just followed an example using some rules showed in a lecture

with online calculators the first step is Factoring (𝑦̅)(𝑧̅)(x+𝑥̅) + (𝑦̅)x(𝑧̅ + z)
and im not sure how that works

[mild tide]

you interpret the formula wrongly
a.b+a.c = a.(b+c)
not
a+a.(b+c)
I've learned this before and I'm sure about it

that is why there's an additional y' in your final result

[autumn siren]

ahh i think i have it now

𝑥.𝑦̅.𝑧 + 𝑥.𝑦̅.𝑧̅ + 𝑥̅.𝑦̅.𝑧̅

𝑥.𝑦̅.(z + 𝑧̅) + 𝑥̅.𝑦̅.𝑧̅ (Distributive Law)
𝑥.𝑦̅.1 + 𝑥̅.𝑦̅.𝑧̅ (Complement Law)
𝑥.𝑦̅ + 𝑥̅.𝑦̅.𝑧̅ (Identity Law )
𝑦̅(𝑥̅.𝑧̅ + x) (Distributive Law)
𝑦̅(𝑧̅ + x) (Absorption Law)
𝑦̅.𝑧̅ + x.𝑦̅ (Distribution )
x.𝑦̅ + 𝑧̅.𝑦̅ (Commutative Law)

il leave this open for a day or so just to see if anyone has more input or i have done this wrong

[flint forge]

we use a ⭐truth table⭐
x y z 1 2 (1->left,2->right)
0 0 0 1 1
0 0 1 do the rest
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

@autumn siren

[autumn siren]

you mean this? #1188024104381583422 message

[flint forge]

yes

[autumn siren]

the point was to do it using the algebraic method

[flint forge]

the absorption law seems weird

i mean, you should add another step there

a+bc = (a+b)(a+c)

[autumn siren]

i think you mean the distributive one above?

[flint forge]

oh you are using the laws used on the previous line.

y'(x'z'+x)
= y'(x'+x)(z'+x)
= y'(z'+x) (used identity law btw)
= xy'+y'z'

[autumn siren]

yeah it was what was used to make it

[autumn siren]

@flint forge so it should be this
𝑥.𝑦̅.(z + 𝑧̅) + 𝑥̅.𝑦̅.𝑧̅ (Distributive Law)
𝑥.𝑦̅.1 + 𝑥̅.𝑦̅.𝑧̅ (Complement Law)
𝑥.𝑦̅ + 𝑥̅.𝑦̅.𝑧̅ (Identity Law )
𝑦̅(𝑥̅.𝑧̅ + x) (Distributive Law)
𝑦̅(𝑥̅+x)(𝑧̅ + x) (Distributive Law)
𝑦̅(𝑧̅ + x) (Identity Law)
𝑦̅.𝑧̅ + x.𝑦̅ (Distribution )
x.𝑦̅ + 𝑧̅.𝑦̅ (Commutative Law)
?

[flint forge]

yes

[autumn siren]

+close