Evaluation of Mathematical Expressions with Sets

What are the evaluations of the given expressions?

Let's break down each evaluation:

Answer:

1. 2|a|: The absolute value of each element in set a is taken and multiplied by 2, resulting in a new set with the absolute values of the original elements multiplied by 2.

2. 2a: Each element in set a is multiplied by 2, resulting in a new set with each element multiplied by 2.

3. 2c: The empty set c, when multiplied by 2, remains as an empty set since there are no elements to multiply.

4. 2d: The set d contains a single element, which is the empty set (∅). When multiplied by 2, the resulting set still contains the empty set twice.

5. |2e|: The set e contains two elements, one of which is the empty set (∅) and the other is a set containing the empty set ({∅}).

Mathematical expressions involving sets can be evaluated in various ways, depending on the operations defined for the sets. In this case, the given expressions involve sets a, b, c, d, and e, each with their own properties.

Evaluation 1: 2|a|

When evaluating 2|a|, we first need to calculate the absolute value of each element in set a. After obtaining the absolute values, we multiply each result by 2 to get the final set as the product. This process ensures that all elements in set a are transformed according to the given operations.

Evaluation 2: 2a

For the expression 2a, it simply involves multiplying each element in set a by 2. This straightforward operation results in a new set where every element is doubled in value compared to the original set a.

Evaluation 3: 2c

When we evaluate 2c, we encounter an interesting scenario with the empty set c. Multiplying the empty set by 2 does not change its essence, leading to the unchanged empty set as the final result.

Evaluation 4: 2d

The set d consists of a single element, which is the empty set (∅). When we multiply this element by 2, we still end up with the empty set repeated twice within the set d.

Evaluation 5: |2e|

Considering the set e with two elements, including the empty set (∅) and a set containing the empty set ({∅}), the absolute value of each element is taken in this evaluation. The resulting set would then be a combination of the absolute values of both elements in set e.