Statements and Quantifiers
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1 Statements and Quantifiers MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018
2 Symbolic Logic Today we begin a study of logic. We will use letters to represent statements symbols for special words such as and, or, and not. Our goal is to determine the truth value of complex statements.
3 Terminology A statement is a declarative sentence that is true or false, but not both simultaneously. A compound statement is formed by combining two or more statements. Logical connectives (words such as and, or, not, and if... then ) are used to form compound statements.
4 Examples Determine if the following sentences are statements. On your i>clicker2 select A: for statement or B: for not a statement. The ZIP code for Millersville, PA is
5 Examples Determine if the following sentences are statements. On your i>clicker2 select A: for statement or B: for not a statement. The ZIP code for Millersville, PA is Yield to oncoming traffic.
6 Examples Determine if the following sentences are statements. On your i>clicker2 select A: for statement or B: for not a statement. The ZIP code for Millersville, PA is Yield to oncoming traffic. Four minus three is two or one plus one is seven.
7 Examples Determine if the following sentences are statements. On your i>clicker2 select A: for statement or B: for not a statement. The ZIP code for Millersville, PA is Yield to oncoming traffic. Four minus three is two or one plus one is seven. A kilogram does not weigh more than a pound.
8 Examples Determine if the following sentences are statements. On your i>clicker2 select A: for statement or B: for not a statement. The ZIP code for Millersville, PA is Yield to oncoming traffic. Four minus three is two or one plus one is seven. A kilogram does not weigh more than a pound. If Tom is a politician, then Jack is a criminal.
9 Negations Every statement has a negation, the statement that is false if the original statement is true.
10 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. Millersville had no snow this winter. All students will take a make-up test. Everyone loves Raymond. x < 5
11 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. The flowers are not blooming. Millersville had no snow this winter. All students will take a make-up test. Everyone loves Raymond. x < 5
12 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. The flowers are not blooming. Millersville had no snow this winter. Millersville had snow this winter. All students will take a make-up test. Everyone loves Raymond. x < 5
13 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. The flowers are not blooming. Millersville had no snow this winter. Millersville had snow this winter. All students will take a make-up test. Not all students will take a make-up test. Everyone loves Raymond. x < 5
14 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. The flowers are not blooming. Millersville had no snow this winter. Millersville had snow this winter. All students will take a make-up test. Not all students will take a make-up test. Everyone loves Raymond. Not everyone loves Raymond. x < 5
15 Negations Every statement has a negation, the statement that is false if the original statement is true. What is the negation of each of the following statements? The flowers are blooming. The flowers are not blooming. Millersville had no snow this winter. Millersville had snow this winter. All students will take a make-up test. Not all students will take a make-up test. Everyone loves Raymond. Not everyone loves Raymond. x < 5 x 5
16 Symbols We will frequently represent statements with letters (p, q, or r) and use special symbols for logical connectives. Connective Symbol Statement Type and Conjunction or Disjunction not Negation
17 Example Consider the following statements: p: Beth has a cold. q: Chris is at the movies. Translate the following symbol statements into words. q p q p q p q
18 Example Consider the following statements: p: Beth has a cold. q: Chris is at the movies. Translate the following symbol statements into words. q p q p q p q Chris is not at the movies.
19 Example Consider the following statements: p: Beth has a cold. q: Chris is at the movies. Translate the following symbol statements into words. q p q p q p q Chris is not at the movies. Beth has a cold or Chris is at the movies.
20 Example Consider the following statements: p: Beth has a cold. q: Chris is at the movies. Translate the following symbol statements into words. q p q p q p q Chris is not at the movies. Beth has a cold or Chris is at the movies. Beth has a cold and Chris is not at the movies.
21 Example Consider the following statements: p: Beth has a cold. q: Chris is at the movies. Translate the following symbol statements into words. q p q p q p q Chris is not at the movies. Beth has a cold or Chris is at the movies. Beth has a cold and Chris is not at the movies. Beth does not have a cold and Chris is not at the movies.
22 Quantifiers Universal quantifiers: words such as all, each, every, no, and none. Existential quantifiers: words such as some, there exists, (for) at least one.
23 Quantifiers Universal quantifiers: words such as all, each, every, no, and none. Existential quantifiers: words such as some, there exists, (for) at least one. We will use the following ideas to form the negation of quantified statements. Statement All do. Some do. Negation Some do not. None do.
24 Example Form the negations of the following statements. All pictures have frames. No picture has a frame. At least one picture has a frame. At least one picture does not have a frame. No picture does not have a frame.
25 Example Form the negations of the following statements. All pictures have frames. Some pictures do not have frames. No picture has a frame. At least one picture has a frame. At least one picture does not have a frame. No picture does not have a frame.
26 Example Form the negations of the following statements. All pictures have frames. Some pictures do not have frames. No picture has a frame. Some picture has a frame. At least one picture has a frame. At least one picture does not have a frame. No picture does not have a frame.
27 Example Form the negations of the following statements. All pictures have frames. Some pictures do not have frames. No picture has a frame. Some picture has a frame. At least one picture has a frame. No picture has a frame. At least one picture does not have a frame. No picture does not have a frame.
28 Example Form the negations of the following statements. All pictures have frames. Some pictures do not have frames. No picture has a frame. Some picture has a frame. At least one picture has a frame. No picture has a frame. At least one picture does not have a frame. All pictures have frames. No picture does not have a frame.
29 Example Form the negations of the following statements. All pictures have frames. Some pictures do not have frames. No picture has a frame. Some picture has a frame. At least one picture has a frame. No picture has a frame. At least one picture does not have a frame. All pictures have frames. No picture does not have a frame. Some picture does not have a frame.
30 Reminder: Sets of Numbers Natural Numbers N = {1, 2, 3, 4,...} Whole Numbers {0, 1, 2, 3, 4,...} Integers Z = {..., 3, 2, 1, 0, 1, 2, 3,...} Rational { Numbers } p Q = q p and q are integers with q 0 Real Numbers R = {x x is a number that can be expressed as a decimal} Irrational Numbers I = {x x is a real number and x cannot be expressed as quotient of integers }
31 Examples Involving Number Sets Determine whether each of the following statements about sets of numbers is true or false. Use your i>clicker2 to select A: for true and B: for false. Every natural number is an integer.
32 Examples Involving Number Sets Determine whether each of the following statements about sets of numbers is true or false. Use your i>clicker2 to select A: for true and B: for false. Every natural number is an integer. There exists a natural number which is not an integer.
33 Examples Involving Number Sets Determine whether each of the following statements about sets of numbers is true or false. Use your i>clicker2 to select A: for true and B: for false. Every natural number is an integer. There exists a natural number which is not an integer. All irrational numbers are real numbers.
34 Examples Involving Number Sets Determine whether each of the following statements about sets of numbers is true or false. Use your i>clicker2 to select A: for true and B: for false. Every natural number is an integer. There exists a natural number which is not an integer. All irrational numbers are real numbers. Each rational number is a positive number.
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