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TEXT IN RED IS INCORRECT
A set is a well-defined collection of objects. The objects are called elements or sometimes members. If A is a set and x is an element of A we write x∈A. If A and B are sets and every element of A is also an element of B we say that A is a subset of B and write A⊂B. There is a unique set having no elements called the empty set and denoted ∅. Notice that ∅⊂A for every set A.
The intersection of two sets is the collection of objects which are elements of both. The intersection of the sets A and B is denoted A∩B. The union of two sets is the collection of all objects which are elements of one or both of them. The union of the sets A and B is denoted A∪B. Clearly, A∩B is both A and B and are subsets of A∪B.
When discussing sets and subsets a universal set is agreed upon. This set contains all objects under discussion. If U is the universal set and A is a subset of U then the complement of A is the collection of elements of U that are not in A. It is denoted AC. Notice that this notion depends on the underlying universal set. The complement of the empty set is the universal set and the complement of the universal set is .
We describe sets by listing their elements inside , using ... when the intent is obvious. For example,
{A, B, C, ... , L}={A, B, C, D, E, F, G, H, I, J, K, L} and {-4, -2, 0, ... , 12}={-4, -2, -0, 2, 4, 6, 8, 10, 12}
are finite sets (those having finitely many elements) while
{1, 2, 3, ...} and {..., -4, -2, 0, 2, 4, ...}={0, ±2, ±4, ...}
are infinite sets. Since a set is a collection rather than a list, order is and no object can be in a set . Thus
{C, A, N, A, D, A}={A, C, D, N}.
Sometimes the truth value of a proposition depends on a variable. The variable is considered to take its values from a specified and the set of all values of the variable which make the proposition true is called the truth set of the proposition.
For example, if we take the universal set to be the set of all trains, "x stops at this station" is a proposition that is true for some values of x and false for other values. The truth set of the proposition is the set of trains that stop at this station.
If U is the universal set and p is a proposition depending on the variable x we use the notation {x∈U : p} to denote the truth set of p. This way of describing sets is also called .
For example, if we take the universal set to be the set N=
There is an important correspondence between the effect of connectives on propositions and the effect of set operations on their truth sets.
Let U denote the universal set and let p and q be propositions that depend on x.