Final answer:
Statement (a) is true: ℘(X∪Y∪Z)⊆℘(X)∪℘(Y)∪℘(Z). To prove this, we consider an element in the left-hand side and show that it is also an element in the right-hand side. Statement (b) is false: (X∩Y)−(Z−X) is not equal to X∩Y.
Step-by-step explanation:
Statement (a) is true: ℘(X∪Y∪Z)⊆℘(X)∪℘(Y)∪℘(Z).
To prove this, let's consider an element, say A, in the left-hand side (℘(X∪Y∪Z)). This means that A is a subset of X∪Y∪Z.
This implies that A must be a subset of X, Y, and Z individually, as any element in X∪Y∪Z must be present in at least one of X, Y, or Z.
Therefore, A is also a subset of ℘(X), ℘(Y), and ℘(Z) individually. Hence, A is an element of ℘(X)∪℘(Y)∪℘(Z).
Since this is true for all elements in the left-hand side, we conclude that ℘(X∪Y∪Z) is indeed a subset of ℘(X)∪℘(Y)∪℘(Z). The laws used in this proof are the definition of the power set and the subset property, which states that if A is a subset of B and B is a subset of C, then A is a subset of C.
Statement (b) is false: (X∩Y)−(Z−X) is not equal to X∩Y.
To demonstrate this, let's define sets X, Y, and Z as follows:
- X = {1,2}
- Y = {2,3}
- Z = {3,4}
Then, we have (X∩Y)−(Z−X) = {2}−{1} = {2}, whereas X∩Y = {2,3}.
Therefore, (X∩Y)−(Z−X) is not equal to X∩Y, and statement (b) is false.