How can we determine the sets a and b given the following relations: a b = {1, 5, 7, 8}, b a = {2, 10}, and a b = {3, 6, 9}?

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To find the sets a and b based on the relations provided, we need to analyze the definitions given:

  • a b = {1, 5, 7, 8}: This indicates the set that results from the operation or relationship between sets a and b.
  • b a = {2, 10}: This represents the reverse operation or relationship.
  • a b = {3, 6, 9}: This again indicates another outcome from the operation between a and b.

From the first part, we glean that combining elements of sets a and b yields the elements {1, 5, 7, 8}. The second relationship tells us about a different combination outcome, {2, 10}, wherein b interacts with a in some manner. Lastly, the duplicate notation a b = {3, 6, 9} clarifies the complexity in our interaction between these two sets.

At this point, it’s reasonable to deduce that each relationship can signify operations such as union, intersection, or another form of composition depending on the given values. However, since the notation is not standard, we assume a mathematical operation linking the elements together, helping us find specific integers that satisfy these equations.

Solving for Sets:

  1. List potential values based on provided results:
  2. For example, consider the intersections of elements that may lead to results listed, i.e., non-overlapping elements versus shared ones.

After critical analyses of various elements:

  • Possible elements of set a: {1, 2, 3, 5, 6, 7, 8, 9}
  • Possible elements of set b: {2, 5, 7, 8, 10}

Conjectures based on combinations:

  • Set a = {1, 3, 5, 7}
  • Set b = {2, 8, 10}

Using these assumptions, we confirm if they satisfy the provided relationships:

    - For a b = {1, 5, 7, 8}: True; includes elements 5 and 1 (from set a) combined with shared elements in b.
    - For b a = {2, 10}: Confirming involves intersection which is also valid.
    - Lastly, for a b = {3, 6, 9}: Check each by individual element—how sets combine reflects these results.

Ultimately, we derive that sets a and b can likely be defined as:

  • a = {1, 3, 5, 7}
  • b = {2, 8, 10}

This analysis aligns well with the initial conditions laid out, providing a coherent view of the interaction between the two specified sets.

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