Given A={1,2,3} and B={2,{3}}. Determine whether each of the following statements is TRUE or FALSE.
a) ∅ C B
b) B C A
c) {3} C A
d) {2,3} C A
e) {2,{3}} ∈ P(B)
f) | P(B) | = 4
g) {0,1} c A
To determine whether each of the statements is true or false, let's go through them one by one:
a) ∅ C B
To determine if the empty set (∅) is a subset of B, we need to check if all the elements of the empty set are also elements of B. Since the empty set has no elements, this statement is true because vacuously all elements of ∅ are in B.
b) B C A
To determine if B is a subset of A, we need to check if all the elements of B are also elements of A. In this case, B has two elements, including a set {3}. Since {3} is not an element of A, this statement is false.
c) {3} C A
To determine if {3} is a subset of A, we need to check if all the elements of {3} are also elements of A. Since 3 is in A, this statement is true.
d) {2,3} C A
To determine if {2,3} is a subset of A, we need to check if all the elements of {2,3} are also elements of A. Since both 2 and 3 are in A, this statement is true.
e) {2,{3}} ∈ P(B)
To determine if {2,{3}} is an element of the power set of B, we need to check if {2,{3}} is one of the subsets of B. The power set of B consists of all possible subsets of B, including B itself. Since {2,{3}} is one of the subsets of B, this statement is true.
f) |P(B)| = 4
To determine the cardinality of the power set of B, we need to count the number of subsets in P(B). In this case, B has two elements, so P(B) will have 2^n subsets, where n is the number of elements in B. Therefore, |P(B)| = 2^2 = 4. This statement is true.
g) {0,1} C A
To determine if {0,1} is a subset of A, we need to check if all the elements of {0,1} are also elements of A. Since neither 0 nor 1 are in A, this statement is false.