Question

Select the statement that is false: group of answer choices 2,3 ⊆ p(a), ∅ ∈ p(a), ∅ ⊆ p(a), 2,3 ∈ p(a)?

1) group of answer choices 2,3 ⊆ p(a)
2) ∅ ∈ p(a)
3) ∅ ⊆ p(a)
4) 2,3 ∈ p(a)

Answer 1

The false statement is '2,3 ⊆ p(a)' because {2,3} is typically an element of the power set p(a), not a subset. The correct statement should be '{2,3} ∈ p(a)'.

Step-by-step explanation:

The student's question is asking to identify the false statement among the ones provided, which concern the properties of sets and the power set, denoted as p(a). To determine which statement is false, we must first understand the notation and the properties of sets and power sets. Here, p(a) refers to the power set of set a, which is the set of all subsets of a, including the empty set ∅ and a itself.

Analysis of Each Statement

2,3 ⊆ p(a): This statement is asserting that the set containing elements 2 and 3 is a subset of the power set of a. This is typically false unless set a itself contains exactly the elements 2 and 3, which we do not know from the given information. Generally, {2,3} is an element of p(a), not a subset.

∅ ∈ p(a): This statement is true, as the empty set is always a member of any power set.

∅ ⊆ p(a): This is also true, as the empty set is a subset of all sets, including power sets.

2,3 ∈ p(a): This statement would be true if set a contains both 2 and 3, making {2,3} a subset of a and thus an element of the power set p(a).

In this case, the first statement is the false one, and the correction would be: {2,3} ∈ p(a).