Final answer:
To prove the distributive property for sets, we need to show that both sets contain the same elements. The left-hand side, a∪(b∩c), contains all the elements that are in either a or in the intersection of b and c. The right-hand side, (a∪b)∩(a∪c), contains all the elements that are in both a∪b and a∪c.
Step-by-step explanation:
To prove the distributive property for sets, a∪(b∩c) = (a∪b)∩(a∪c), we need to show that both sets contain the same elements.
Let's start with the left-hand side, a∪(b∩c).
This set contains all the elements that are in either a or in the intersection of b and c. In other words, it contains all the elements of a, plus any elements that are in both b and c.
Now let's look at the right-hand side, (a∪b)∩(a∪c).
This set contains all the elements that are in both a∪b and a∪c. In other words, it contains all the elements that are in both a or b, and also in both a or c.
Since both sets contain the same elements, we can conclude that the distributive property for sets holds true: a∪(b∩c) = (a∪b)∩(a∪c).