Let U={1,2,3,4,5,6,7,8,9,10} A={1,3,5,7,9} B={1,2,4,7,8} C={2,4,6,8} Verify

step 1 Universal set is given 1 .2 .3 .4 .5. 6 .7 .8 .9. 9. And given equal to 2 .4 .6 .6 .8. And set b

Let U={1,2,3,4,5,6,7,8,9,10} A={1,3,5,7,9} ...

Let
$$
\begin{array}{l}
U=\{1,2,3,4,5,6,7,8,9,10\} \\
A=\{1,3,5,7,9\} \\
B=\{1,2,4,7,8\} \\
C=\{2,4,6,8\}
\end{array}
$$
Verify each equation by direct computation.
a. $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
b. $(A \cup B)^{c}=A^{c} \cap B^{c}$

Key Concepts

Verification by Direct Computation

Verification by direct computation involves methodically listing and examining each element in the sets involved in an identity or equation. This approach ensures that both sides of the equation yield identical sets by comparing their elements, thereby confirming the validity of set relationships and laws.

De Morgan's Laws

De Morgan's Laws formalize the relationship between unions and intersections through complementation, stating that the complement of a union is equivalent to the intersection of the complements, and the complement of an intersection is equivalent to the union of the complements. These laws are essential for translating between different forms of set expressions, such as (A?B)^c = A^c ? B^c.

Distributive Laws

The distributive laws in set theory reveal how intersection distributes over union (and vice versa), allowing the expression of one set operation in terms of others. This property is crucial for simplifying and rearranging complex set expressions, providing a pathway to verify identities like A?(B?C) = (A?B)?(A?C).

Set Operations

Set operations such as union, intersection, and complement are fundamental concepts that enable the combination and comparison of sets. The union of sets gathers all unique elements from each, the intersection finds common elements shared by sets, and the complement identifies elements in a universal set that are not in a specified subset.

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