3. SETS AND SET OPERATIONS
1. 3.1 SETS, ELEMENTS, AND NOTATION
What we understand a set to be is a collection of objects (called elements
of the set) which are characterized by some property that lets us think of the
collection as a unit. The property
1.11 UNIVERSAL AND EXISTENTIAL QUANTIFIERS.
In section 1.9, one of the examples was If a person is a Texan, then he is an American.
We could have also said Every Texan is an American and have meant the same thing.
The latter statement is a still a conditi
1.9 RELATED CONDITIONAL STATEMENTS.
There are three other conditional statements related to the conditional If p, then q. These
are the converse If q, then p, the inverse If not p, then not q, and the
contrapositive If not q, then not p. In writing the
1.7. CONDITIONAL STATEMENTS.
The statement If p, then q. is known as a conditional statement. The statement p is
called the hypothesis, and the statement q is called the conclusion. The symbol for a
conditional statement is p q, which may also be re
1.5 LOGICALLY EQUIVALENT STATEMENTS, TAUTOLOGIES, &
Two statements p and q are logically equivalent provided that they have exactly the
same truth values in all possible cases; i.e., their truth table columns are identical.
1.12 DENIALS OF STATEMENTS WITH UNIVERSAL AND EXISTENTIAL
In order for the statement (" x) (P(x) to be false, the solution set S can not be U. Hence
there must be at least one element of U that is not in S; that is, there is an element x o
2. METHODS OF PROOF
In every mathematical system, there two types of terms: defined and undefined.
Examples of undefined terms are set, element, in, number, point, between, and line. Then
there are terms which are defined in terms of the
3.7. DISPROVING A CONDITIONAL STATEMENT
The statement ("x)(P(x)is false when the statement ($x)( P(x) is true. One SPECIFIC
exampleofanx which makes P(x) false is called a counterexample, and giving a
counterexample is called disproving the statement
*Theorem 1. If A is a set, then .1) AA
.2) AA=A .3) AA=A
8. If A and B are sets, then A B if and only if
9. If A and B are sets, then x B \ A if and only if
10. Two sets A and B are not disjoint if and only if
3. 3.4. VENN DIAGRAMS
In a given problem we may assume that all of the sets we are discussing are
subsets of some universal set U.
Suchpicturesarereferred to as Venn Diagrams. B
2.3. PROOF BY THE CONTRAPOSITIVE.
This is actually also a direct proof, just not of the given statement, but instead of the
equivalent statement called its contrapositive.
Theorem. p r ( (r p).
Proof: Assume r. Then q.
1.2 DENIALS (also referred to as negations or bare denials).
When we want to refer to statements symbolically, we usually use letters such as p, q, or
If p is a statement, then the denial of p is a statement, denoted by p and read not p,
with the p