Final answer:
The equality of the boolean expressions yz + xyz' + x'y'z and xy + x'z is proven using a truth table by showing that for every combination of x, y, and z values, both expressions yield the same result, confirming they are equivalent and represent a valid deductive inference.
Step-by-step explanation:
To prove the boolean expression yz + xyz' + x'y'z equals xy + x'z using a truth table, we need to evaluate the expressions for all combinations of the boolean variables x, y, and z. Each variable can either be true (1) or false (0). We will compute the value of both expressions for each possible combination of x, y, and z to show that they produce the same result every time, which proves the two expressions are equivalent.
Here's the truth table showing all possible combinations:
xyzyzxyz'x'y'zLeft Expression
(yz + xyz' + x'y'z)xyx'zRight Expression
(xy + x'z)Equivalent?0000000000Yes
Upon completing the table, you will note that the 'Left Expression' and 'Right Expression' columns always match, which means the expressions are indeed equivalent. Hence, the initial hypothesis is correct and can be considered a valid deductive inference.