Final answer:
To confirm that X.(Y+Z) equals (X.Y)+(X.Z), a truth table can be constructed to evaluate all possible truth values for X, Y, and Z. The truth table shows that the statement holds in all scenarios, thus verifying the distributive law in Boolean algebra.
Step-by-step explanation:
To determine that the expression X.(Y+Z)=(X.Y)+(X.Z) is true, we can use a truth table to evaluate all possible combinations of truth values for the variables X, Y, and Z.
- There are three variables involved: X, Y, and Z. Each can either be true (T) or false (F), which makes for 2^3 or 8 different combinations of truth values.
- To construct the truth table, list all possible combinations of truth values for X, Y, and Z.
- Then calculate the truth value for Y+Z (logical OR), and X.(Y+Z) (logical AND).
- Separately calculate the truth values for X.Y and X.Z (both logical ANDs), and then the value of (X.Y)+(X.Z) (logical OR).
- Compare the results of X.(Y+Z) with (X.Y)+(X.Z) for each row. If they match on all rows, the expression is proved to be true.
Conclusion
Using the truth table, we see that the original statement X.(Y+Z)=(X.Y)+(X.Z) holds true in every possible scenario, which confirms that the law is valid. This is an example of the distributive law in Boolean algebra.