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November 11th, 2017, 05:28 AM  #1 
Member Joined: Oct 2017 From: Rumba Posts: 39 Thanks: 0  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 
November 11th, 2017, 08:48 AM  #2  
Senior Member Joined: Aug 2012 Posts: 2,305 Thanks: 705  Quote:
No, the empty set is a subset of every set, so this is true. No. What is $P(A)$? The power set of the empty set is $\{\emptyset\}$, which is an element of B. This is true. No, this is false. It is not the case that every element of B is an element of A. Yes. The empty set is a subset of every set. Yes. Every set is a subset of itself. Last edited by Maschke; November 11th, 2017 at 08:51 AM.  
November 11th, 2017, 09:00 AM  #3 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,150 Thanks: 875 Math Focus: Wibbly wobbly timeywimey stuff. 
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 
November 11th, 2017, 09:47 AM  #4  
Senior Member Joined: Aug 2012 Posts: 2,305 Thanks: 705  Quote:
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. Last edited by Maschke; November 11th, 2017 at 09:57 AM.  
November 11th, 2017, 05:13 PM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
The difference between (a) and (f) is the difference between "$\displaystyle \in$" and "$\displaystyle \subset$"


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