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1. Given the Venn diagram, write the expression modeled by the blue area in Set B. (include the light blue and the dark blue regions) 2. Given the relationship: (An B)(Anc) a, Simplify the expression. Provide a rule for each simplification step you take. b. Provide a Venn diagram, shading the region that shows the relationship

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Answer #1

The required region represents all the elements of the set B that are not present in the set A Therefore, the required expression is as follows Apply the De-Morgans Law on the above expression to obtain the following expression: B (A B) Since, Intersection is distributive over the union (Distributive Law), therefore, the above expression can be written as follows Since, the intersection of any set with its complement results in a NULL set (Complement Law), therefore, the above expression can be written as follows Since, the union of any set with the NULL set results in the set itself (Identity Law), therefore, the above expression can be written as follows Hence, the required expression is (BnA) The given relationship is as follows Apply the De-Morgans Law on the above expression to obtain the following expression: Apply the Associative Law on the above expression to obtain the following expression: (AnBnAnc) Since, the intersection of a set with itself results in the original set itself (Idempotent Law), therefore, the above expression can be written as follows AnBnC) Hence, the simplified expression is as follows: (AnBnC)b.

The Venn diagram for the relationship provided in part a. is as follows:

In the above Venn Diagram, the sets A, B and, C are the given sets and U is the Universe. The part of the diagram shaded in green represents the area represented by the relationship in the part a.

Note: If the universe is not being considered and only the region of the sets A, B and, C are to be considered then the required Venn diagram is as follows:

In the above Venn diagram, the sets A, B and, C are the given sets and the part shaded in green represents the area represented by the relationship given in part a.

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