In this problem, we prove the following statement: Let A, B be sets in a universe U. Then An B C (AUB) Your task is to provide justification for some of the steps. Possible answers are given below. Put a number into each blank (with no parentheses). (1) definition of union (2) definition of intersection (3) definition of subset (4) definition of set subtraction (i.e. S-T) (5) definition of complement (6) DeMorgan's law of logic Proof: Let a e An B. Then ze A and a B by justification 2 4. Since z A, this means that a A by justification 5 Using the same reasoning, x E B means that x B, so (x B) is true. Combining these facts, -(a e A) A-(z B) is true. Therefore, -((A) V (z e B)) is true using justification 6 In other words, (z E AUB) is true using justification 4 Since z AUB, this means that ze (AUB) using justification 3. This concludes the proof that An BC (AUB). D So(A) is true. which is equivalent to AUB.