answer:To define the `plus` function, we can use pattern matching and recursion. Here's the definition of `plus`: ``` plus : Nat -> Nat -> Nat plus Zero n = n plus (Succ m) n = Succ (plus m n) ``` Now, let's evaluate `plus (Succ (Succ Zero)) (Succ Zero)`: ``` plus (Succ (Succ Zero)) (Succ Zero) = Succ (plus (Succ Zero) (Succ Zero)) -- by the second clause of the plus definition = Succ (Succ (plus Zero (Succ Zero))) -- by the second clause of the plus definition again = Succ (Succ (Succ Zero)) -- by the first clause of the plus definition ``` So the result is `Succ (Succ (Succ Zero))`, which represents the natural number 3.,We can define the `plus` function as follows: ``` plus :: Nat -> Nat -> Nat plus Zero n = n plus (Succ m) n = Succ (plus m n) ``` Now, let's evaluate `plus (Succ (Succ Zero)) (Succ Zero)`. ``` plus (Succ (Succ Zero)) (Succ Zero) = plus (Succ Zero) (Succ Zero) = Succ (plus Zero (Succ Zero)) = Succ (Succ Zero) = Succ (Succ Zero) ``` So, the result is `Succ (Succ Zero)`., To define the `plus` function, we can use pattern matching to handle the different cases for the input natural numbers. Here's the implementation: ``` plus : Nat -> Nat -> Nat plus Zero n = n plus (Succ m) n = Succ (plus m n) ``` Now let's evaluate `plus (Succ (Succ Zero)) (Succ Zero)` using the provided function: ``` plus (Succ (Succ Zero)) (Succ Zero) = { plus (Succ m) n = Succ (plus m n) } Succ (plus (Succ Zero) (Succ Zero)) = { plus (Succ m) n = Succ (plus m n) } Succ (Succ (plus Zero (Succ Zero))) = { plus Zero n = n } Succ (Succ (Succ Zero)) ``` So the result is `Succ (Succ (Succ Zero))`, which represents the number 3.

answer:The intersection, or the set containing the elements that are common to both sets, is: A ∩ B = {3, 4, 5}, The intersection of two sets A and B is the set of elements that belong to both A and B. To find the intersection of sets A and B, we need to identify the common elements in both sets: A ∩ B = {x | x ∈ A and x ∈ B} In this case, the common elements are 3, 4, and 5. Therefore: A ∩ B = {3, 4, 5},The intersection of the two sets A and B is the set of elements that are common to both A and B. In this case, the intersection is: A ∩ B = {3, 4, 5}

answer:The cardinality of a set is the number of distinct elements in the set. In the set A = {1, 2, {3, 4}, {1, 2}}, there are three distinct elements: 1, 2, and {3, 4}. The set {1, 2} is not counted as a distinct element because it contains the same elements as the set itself, so it is not considered a separate element. Therefore, the cardinality of the set A is 3.,The cardinality of a set is the number of elements in the set. In this case, set A = {1, 2, {3, 4}, {1, 2}} has 4 elements: 1, 2, {3, 4}, and {1, 2}. Since two of the elements (3 and 4) are inside the other element ({3, 4}), it is a situation of overlapping elements or elements within other elements. To accurately calculate the cardinality in this situation, you need to count the distinct elements only once, regardless of whether they are parts of other elements. Therefore, the cardinality of A is 3., The cardinality of a set is the number of elements it contains. In this case, set A contains four distinct elements: 1, 2, {3, 4}, and {1, 2}. Notice that {3, 4} and {1, 2} are themselves sets, so they are considered individual elements in set A. Therefore, the cardinality of set A is 4.

answer:The intersection of two sets A and B is the set of elements that are common to both A and B. In this case, the intersection of A and B is: A ∩ B = {2, 4}, The intersection of two sets A and B is the set of elements that are common to both sets. In this case, the set A contains the elements 1, 2, 3, 4, and 5, while the set B contains the elements 2, 4, and 6. The common elements between sets A and B are 2 and 4. Therefore, the intersection of sets A and B is: A ∩ B = {2, 4},The intersection of sets A and B contains all elements that are present in both sets. In this case, the intersection is: {2, 4}