flyingpig said:C is a subset of D
if x is in the intersection of A and B, then x belongs to both A and B
SammyS said:∴ x belongs to A .
[tex]x \in A \cap B \iff x \in A [/tex] and [tex] x \in B[/tex]
[tex]\therefore x \in A[/tex]
So [tex]A \cap B \subseteq A[/tex]
[tex]x \in A \cup B \iff x \in A[/tex] or [tex]x \in B[/tex]
So [tex]A \subseteq A \cup B[/tex]
q.e.d
A set is a collection of distinct objects, called elements, that are grouped together based on a common characteristic or property.
To prove that two sets are equal, you need to show that they have the same elements. This can be done by listing out the elements of both sets and comparing them, or by using set notation and proving that all elements in one set belong to the other set and vice versa.
Yes, you can prove that a set is empty by showing that it has no elements. This can be done by listing out the elements of the set and showing that there are none, or by using set notation and proving that no elements belong to the set.
To prove that an element belongs to a set, you can either show that the element is explicitly listed as an element of the set or that it satisfies the defining property of the set. For example, if the set is defined as all even numbers, you can prove that 4 is an element of the set by showing that it is an even number.
A subset is a set that contains all the elements of another set. A proper subset is a subset that contains some, but not all, of the elements of another set. In other words, a proper subset is a subset that is not equal to the original set.