6.1 Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced mathematical methods to logic in 1847 while Georg Cantor did theoretical work on sets and discovered that there are many different sizes of infinite sets. Statements or Propositions A proposition or statement is a declaration which is either true or false. Some examples: 2+2 = 5 is a statement because it is a false declaration. Orange juice contains vitamin C is a statement that is true. Open the door. This is not considered a statement since we cannot assign a true or false value to this sentence. It is a command, but not a statement or proposition. Negation The negation of a statement, p , is “not p” and is denoted by ┐ p Truth table: p ┐ p TF FT If p is true, then its negation is false. If p is false, then its negation is true. 1 Disjunction A disjunction is of the form p V q and is read p or q. Truth table for disjunction: p q p V q TT T TF T FT T FF F A disjunction is true in all cases except when both p and q are false. Conjunction A conjunction is only true when both p and q are true. Otherwise, a conjunction of two statements will be false: Truth table: p q p ∧ q TT T TF F FT F FF F Conditional statement To understand the logic behind the truth table for the conditional statement, consider the following statement. “If you get an A in the class, I will give you five bucks.” Let p = statement “ You get an A in the class” Let q = statement “ I will give you five bucks.” Now, if p is true (you got an A) and I give you the five bucks, the truth value of p q is true. The contract was satisfied and both parties fulfilled the agreement. Now, suppose p is true (you got the A) and q is false (you did not get the five bucks). You fulfilled your part of the bargain, but weren’t rewarded with the five bucks. So p q is false since the contract was broken by the other party. Now, suppose p is false. You did not get an A but received five bucks anyway. (q is true) No contract was broken. There was no obligation to receive 5 bucks, so truth value of p q cannot be false, so it must be true. Finally, if both p and q are false, the contract was not broken. You did not receive the A and you did not receive the 5 bucks. So p q is true in this case. 2 Truth table for conditional p q p q TT T TF F FT T FF T Variations of the conditional Converse: The converse of p q is q p Contrapositive: The contrapositive of p q is ┐q ┐p Examples Let p = you receive 90% Let q = you receive an A in the course p q ? If you receive 90%, then you will receive an A in the course. Converse: q p If you receive an A in the course, then you receive 90% Is the statement true? No. What about the student who receives a score greater than 90? That student receives an A but did not achieve a score of exactly 90%. 3 Example 2 State the contrapositive in an English sentence: Let p = you receive 90% Let q = you receive an A in the course p q ? If you receive 90%, then you will receive an A in the course ┐q ┐p If you don’t receive an A in the course, then you didn’t receive 90%. The contrapositive is true not only for these particular statements but for all statements , p and q. Logical equivalent statements Show that pq → is logically equivalent to ¬∨p q We will construct the truth tables for both sides and determine that the truth values for each statement are identical. The next slide shows that both statements are logically equivalent. The red columns are identical indicating the final truth values of each statement. 4 6.2 Sets This section will discuss the symbolism and concepts of set theory Set properties and set notation Definition of set: A set is any collection of objects specified in such a way that we can determine whether or not an object is or is not in the collection. Example 1. Set A is the set of all the letters in the alphabet. Notation: A = { a, b, c, d, e, …z) We use capital letters to represent sets. We list the elements of the set within braces. The three dots … indicate that the pattern continues. We can determine that an object is or is not in the collection. For example . e ∈ A stands for “e is an element of , or e belongs to set A” . This statement is true. The statement “ 3 ∈ A is false, since the number 3 is not an element of set A. The statement “ 3 A”is true. Null set Example. What are the real number solutions of the equation? 2 x +=10 Answer: There are no real number solutions of this equation since no real number squared added to one can ever equal 0. We represent the solution as the null set { } or Ø 1 Set builder notation Sometimes it is convenient to represent sets using what is called set-builder notation. For example, instead of representing the set A, letters in the alphabet by the roster method, we can use set builder notation: {x x is letter of the English alphabet} means the same as { a , b , c, d, e , …z} 2 Example two. { x l x = 9 } = {3 , -3} . This is read as the set of all such that the square of x equals 9. The solution set consists of the two numbers 3 and -3. Subsets A ⊂ B means A is a subset of B. A is a subset of A if every element of A is also contained in B. For example, the set of integers denoted by { …-3, -2, -1, 0, 1, 2, 3, …} is a subset of the set of real numbers. Formal definition of subset: A ⊂ B means if x ∈ A, the x ∈ B ∅ (null set )is a subset of every set. To verify this statement, let’s use the definition of subset. “ if x ∈ ∅ , then x is an element of A. But since the null set contains no elements, the statement x is an element of the null set is false. Hence, we have a conditional statement in which the premise is false. We know that p q is true if p is false. Since p is false, we conclude that the conditional statement is true. That is “ if x belongs to the null set, then x belongs to set A” is true, which implies that the null set must be a member of every set. Therefore, the null set is a subset of every set. Number of subsets List all the subsets of set A = {bird, cat, dog} For convenience, we will use the notation A = {b , c, d} to represent set A. Solution: ∅ is a subset of A. We also know that every set is a subset of itself so A = {b , c, d } is a subset of set A since every element of set A is contained within set A. How many two-element subsets are there? We have {b, c}, {b, d} , {c, d} How many one-element subsets? { b} , {c} and {d} . There is a total of 8 subsets of set A if you count all the listed subsets. 2 Set operations The union of two sets is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. The symbolism used is The Venn Diagram representing the union of A and B is the entire region shaded yellow. A ∪ B A BxxAorxB=∈{} ∈ Example of Union The union of the rational numbers with the set of irrational numbers is the set of real numbers. Rational numbers are those numbers that can be expressed as fractions, while irrational numbers are numbers that cannot be represented exactly as fractions, such as 2 Rational numbers a/b, ¾, 2/3 , 0.6 Irrational numbers such as square root of two, Pi , square root of 3 Real numbers: represented by shaded blue-green region Intersection of sets A and B The intersection of sets A and B is the set of elements that is common to both sets A and B. It is symbolized as A∩ B = { x l x A and x B } A B Represented by Venn Diagrams: Intersection 3 Complement of a set To understand the complement of a set, we must first define the universal set. The set of all elements under consideration is called the universal set. For example, when discussing numbers, the universal set may consist of the set of real numbers. All other types of numbers (integers, rational numbers, irrational numbers ) are subsets of the universal set of real numbers. Complement of set A: The complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism for the complement of set A follows: A’ = { x U|x A} Venn Diagram for complement of set A Yellow region= all The complement of set A is elements in U that represented by the regions are neither in A or B.