True or false. [A �¿ (B �¾ C)]c = (A �¿ B)c �¿ (A �¿ C)c

To determine if the statement is true or false, we can use the laws of set theory to simplify both sides of the equation and see if they are equal.

Let's break this down step by step:

1. Start with the left-hand side: [A �¿ (B �¾ C)]c

2. Apply De Morgan's law to the expression inside the brackets, which states that the complement of the union of two sets is equal to the intersection of their complements. This gives us:

[A �¿ (B �¾ C)]c = [A �¿ (Bc �¿ Cc)]c

3. Using the distributive law, we distribute the intersection operator over the union inside the brackets:

[A �¿ (Bc �¿ Cc)]c = (A �¿ Bc) �¿ (A �¿ Cc)]c

4. Apply De Morgan's law again, this time to the expression inside the brackets on the right-hand side:

(A �¿ Bc) �¿ (A �¿ Cc)]c = (A �¿ Bc)c �¿ (A �¿ Cc)c

5. Applying the complement operation twice in a row gives us:

(A �¿ Bc)c �¿ (A �¿ Cc)c = (A �¾ B) �¿ (A �¾ C)

6. Simplify further using De Morgan's law, which states that the complement of the intersection of two sets is equal to the union of their complements:

(A �¾ B) �¿ (A �¾ C) = (A �¿ Bc) �¿ (A �¿ Cc)

7. Finally, we can apply another De Morgan's law to the right-hand side to get:

(A �¿ Bc) �¿ (A �¿ Cc) = [(A �¿ B) �¿ (A �¿ C)]c

Now we can see that the left-hand side [A �¿ (B �¾ C)]c is equal to the right-hand side [(A �¿ B) �¿ (A �¿ C)]c. Therefore, the statement is true.

In summary, we used De Morgan's law and distributive law to simplify both sides of the equation and showed that they are equal, resulting in a true statement.