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November 11th, 2017, 06:28 AM   #1
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Set notation with empty set

Let A = ∅, B = {7, 8, 9, 1, {∅}}. For each of the following, state whether the statement is
True or False.

(a) ∅ ∈ A = true
(b) A ⊂ B = false
(c) P(A) ∈ B = false
(d) B ⊆ A = true
(e) A ⊆ B = true
(f) A ⊆ ∅ = false
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November 11th, 2017, 09:48 AM   #2
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Let A = ∅, B = {7, 8, 9, 1, {∅}}. For each of the following, state whether the statement is
True or False.

(a) ∅ ∈ A = true
No, the empty set is empty. It does not have any elements. This is false.

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(b) A ⊂ B = false
No, the empty set is a subset of every set, so this is true.

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Originally Posted by sita View Post
(c) P(A) ∈ B = false
No. What is $P(A)$? The power set of the empty set is $\{\emptyset\}$, which is an element of B. This is true.

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Originally Posted by sita View Post
(d) B ⊆ A = true
No, this is false. It is not the case that every element of B is an element of A.

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Originally Posted by sita View Post
(e) A ⊆ B = true
Yes. The empty set is a subset of every set.

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Originally Posted by sita View Post
(f) A ⊆ ∅ = false
Yes. Every set is a subset of itself.
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Last edited by Maschke; November 11th, 2017 at 09:51 AM.
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November 11th, 2017, 10:00 AM   #3
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I don't want to distract from the OP's problem, but I'm curious.... Don't a) and f) contradict each other? If $\displaystyle A \in \emptyset$ because every set is a subset of itself, then why isn't $\displaystyle \emptyset \in A$ by the same argument?

-Dan
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November 11th, 2017, 10:47 AM   #4
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I don't want to distract from the OP's problem, but I'm curious.... Don't a) and f) contradict each other? If $\displaystyle A \in \emptyset$ because every set is a subset of itself, then why isn't $\displaystyle \emptyset \in A$ by the same argument?

-Dan
The empty set is empty, so $x \in \emptyset$ is false for every $x$.

But the empty set is a subset of every set, including itself. We say $X \subset Y$ if it is the case that $x \in X \implies x \in Y$. For any set $X$, it's vacuously true that if $x \in \emptyset$ then $x \in X$, because there is no $x$ that's an element of $X$.

In other words the statement $x \in \emptyset \implies x \in X$ is true, because the antecedent is always false. $x \in \emptyset$ is always false, making the implication true. So by definition, $\emptyset \subset X$.

So the empty set contains no elements, but the empty set is a subset of every set.
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Last edited by Maschke; November 11th, 2017 at 10:57 AM.
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November 11th, 2017, 06:13 PM   #5
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The difference between (a) and (f) is the difference between "$\displaystyle \in$" and "$\displaystyle \subset$"
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