Question

1. Use algebraic manipulation to prove that x + yz = (x + y) • (x + z). Note that this is the

distributive rule, as stated in identity 12b in Section 2.5
2. Use algebraic manipulation to prove that Note that this is the
consensus property 17a in Section 2.5
3. Use algebraic manipulation to prove that
4. Use algebraic manipulation to prove that

Answer 1

The distributive rule
`(x + yz) = (x + y) • (x + z)`, we can start by expanding the right side of the equation using the distributive property.
`(x + y) • (x + z) = x(x + z) + y(x + z)`

Now, we can further simplify this expression by applying the distributive property again.
`= x^2 + xz + xy + yz` Notice that the terms
`xz and xy + yz` are the same as the terms in the left side of the equation (x + yz). Therefore, we can conclude that
`x + yz = (x + y) • (x + z),` which proves the distributive rule.

To prove the consensus property, we need to show that
`(x + y)(x + y') = x.` Using the distributive property, we can expand the left side of the equation.
`(x + y)(x + y') = x(x + y') + y(x + y')` Next, we can simplify this expression by applying the distributive property again.
`= x^2 + xy' + xy + yy'`

Now, notice that xy' + xy is equal to x(y' + y), and since y' + y equals 1 (based on the consensus property), we can simplify the expression further.
`= x^2 + x(1) + yy'`Finally, we can simplify this expression even more by recognizing that x(1) equals x, and yy' equals 0.
`= x + 0`

Thus, we have shown that
`(x + y)(x + y') = x,`which proves the consensus property. (Explanation for the third question is missing) . (Explanation for the fourth question is missing) I apologize, but it seems that the explanations for the third and fourth questions are missing from your request.

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