Correct. You can also express this positively. ##J\subseteq A## means all elements of ##J## are also elements in ##A##. This is true, because all statements about the elements of the empty set are true, or as I like to say: all elements of the empty set have purple eyes.Mr Davis 97 said:Homework Statement
Let ##J = \emptyset## and ##A## be any set. Then ##J \subseteq A##.
Homework Equations
The Attempt at a Solution
Proof: Suppose that it is false that ##J \subseteq A##. Then ##\exists x (x \in J ~~\text{and}~~ x \not \in A)## is true. But this is a contradiction, since ##\exists x (x \in J ~~\text{and}~~ x \not \in A)## is false also, since there exists no element in the empty set.
One slightly tangential question. By definition, ##A \subseteq B## means that ##\forall x (x \in A \implies x \in B)##. But for the latter statement, what exactly is the set that ##x## is quantified over?fresh_42 said:Correct. You can also express this positively. ##J\subseteq A## means all elements of ##J## are also elements in ##A##. This is true, because all statements about the elements of the empty set are true, or as I like to say: all elements of the empty set have purple eyes.
What do you mean? "For all ##x##" is the same as "given any ##x##" with emphasis on any. But as soon as the quantifier is left, ##x## becomes a certain element, because ##x\in A## deals with only one ##x##. However, it cannot be further quantified, since it might not exist.Mr Davis 97 said:One slightly tangential question. By definition, ##A \subseteq B## means that ##\forall x (x \in A \implies x \in B)##. But for the latter statement, what exactly is the set that ##x## is quantified over?
My question is what are the x's under consideration here? Where do they live?fresh_42 said:What do you mean? "For all ##x##" is the same as "given any ##x##" with emphasis on any. But as soon as the quantifier is left, ##x## becomes a certain element, because ##x\in A## deals with only one ##x##. However, it cannot be further quantified, since it might not exist.
Nobody knows. They can be anywhere without having anything to do with ##A## or ##B##. But as soon as one of them is in ##A##, then it has to be also in ##B##. There is no statement about the elements of ##C##. However, all elements of ##C## which happen to be in ##A##, too, have to be as well in ##B##. Same for the sets ##D,E,\ldots ## or whatever - even none.Mr Davis 97 said:My question is what are the x's under consideration here? Where do they live?
The empty set is a mathematical concept that represents a set with no elements. It is denoted by the symbol ∅ or {}. It is different from a set with no elements, which is represented by the symbol {∅}.
To show that the empty set is a subset of every set, we need to prove that every element in the empty set is also an element of the set in question. Since the empty set has no elements, this condition is automatically satisfied and therefore, the empty set is a subset of every set.
Proving that the empty set is a subset of every set is important because it is a fundamental concept in set theory. It helps us to understand the relationship between sets and how they can be compared and related to one another.
No, the empty set cannot be a proper subset of a set. A proper subset is a subset that contains fewer elements than the original set. Since the empty set has no elements, it cannot have fewer elements than any other set and therefore, cannot be a proper subset.
The concept of the empty set can be applied to real-life situations in various ways. For example, in a grocery store, the empty set can represent a shopping cart with no items in it. In a soccer team, the empty set can represent the number of goals scored by a player who did not play in the game. In both cases, the empty set is used to represent the absence of something.