To answer these questions, you need to use some set operations and notation. Here are some definitions and examples:
- The union of two sets A and B, written A ∪ B, is the set of all elements that are in either A or B. For example, {1, 2, 3} ∪ {2, 4, 5} = {1, 2, 3, 4, 5}.
- The intersection of two sets A and B, written A ∩ B, is the set of all elements that are in both A and B. For example, {1, 2, 3} ∩ {2, 4, 5} = {2}.
- The complement of a set A with respect to a universal set U, written Ac or U - A, is the set of all elements in U that are not in A. For example, if U = {1, 2, 3, 4, 5}, then {1, 2, 3}c = {4, 5}.
- The difference of two sets A and B, written A - B or A \ B, is the set of all elements in A that are not in B. For example, {1, 2, 3} - {2, 4, 5} = {1, 3}.
- The Cartesian product of two sets A and B, written A × B, is the set of all ordered pairs (a,b) where a ∈ A and b ∈ B. For example, {1, 2} × {a,b} = {(1,a), (1,b), (2,a), (2,b)}.
- The power set of a set A, written P(A), is the set of all subsets of A. For example, P({1, 2}) = {∅,{1},{2},{1,2}}.
Now let's use these definitions to answer your questions. Let A = {{1},2,3} and B = {a,b,c}. Then:
1. (AUB) CA = ({1},2,a,b,c) - {{1},2} = (a,b,c)
2. AC (AUB) = {{1},3} ∩ ({1},2,a,b,c) = {{1},3}
3. P(A) = P(B) means that the power sets of A and B are equal. This is false because P(A) has eight elements while P(B) has only four elements.
4. (A × B) ≤ P(A × B) means that the Cartesian product of A and B is a subset of the power set of A × B. This is true because every element of A × B is also a subset of itself and hence belongs to P(A × B).
5. (A × B) ≤ P(A) × P(B) means that the Cartesian product of A and B is a subset of the Cartesian product of the power sets of A and B. This is false because there are some elements in P(A) × P(B) that are not in A × B. For example, ({∅},{a}) ∈ P(A) × P(B) but not in A × B.
To prove one of these statements formally, we need to use some logical rules and axioms. I will prove statement 4 as an example.
Proof: Let x be an arbitrary element of A × B. Then x = (a,b) for some a ∈ A and b ∈ B. By definition of subset, x ∈ x. By definition of power set, x ∈ P(A × B). Therefore x ∈ (A × B) implies x ∈ P(A × B). Since x was arbitrary, this holds for any element of A × B. Hence (A × B) ≤ P(A × B). QED.