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Section 1.2. SYMBOLIC LOGIC. Statements. This section we will study symbolic logic which was developed in the late 17 th century. All logical reasoning is based on statements. A statement is a sentence that is either true or false. Which of the following are statements?.

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SYMBOLIC LOGIC

Statements

- This section we will study symbolic logic which was developed in the late 17th century.
- All logical reasoning is based on statements.
- A statement is a sentence that is either true or false.

Which of the following are statements?

- The 2004 Summer Olympic Games were in Athens, Greece.(Statement - true)
- Seinfeld was the best TV comedy of all time. (Not a Statement – opinion)
- Did you watch The Godfather? (Not a statement – a question)
- The Philadelphia Eagles won Super XV. (statement – false)
- I am telling a lie. (not a statement – paradox)

Statements

- Traditionally, symbolic logic uses lower case letters to denote statements. Usually the letters p, q, r, s, t.
- Statements get labels.
- p: It is raining.

Compound Statements

- A compound statement is a statement that contains one or more simpler statements.
- Compound statements can be formed by
- inserting the word NOT,
- joining two or more statements with connective words such as AND, OR, IF…THEN, ONLY IF, IF AND ONLY IF.

Examples

- Steve did not do his homework.
- This is formed from the simpler statement, Steve did his homework.

- Mr. D wrote the MAT114 notes and listened to a Pink Floyd CD.
- This statement is formed from the simper statements: Mr. D wrote the MAT114 notes.
Mr. D listened to a Pink Floyd CD.

- This statement is formed from the simper statements: Mr. D wrote the MAT114 notes.
- Compound statements are known as negations, conjunctions, disjunctions, conditionals or combinations of each.

NEGATION ~p

- The negation of a statement is the denial of that statement. The symbolic representation is a tilde ~.
- Negation of a simple statement is formed by inserting not.
- Example: The senator is a Republican.
The negation is: The senator is not a Republican.

Negation

- “All of Mr. D’s students are Philadelphia Eagles fans.”
- The negation is: “Some of Mr. D’s students are not Philadelphia Eagles fans.”
- To negate the first statement, we don’t need to have all the students to be not Eagles fans, we just need only one student not to be an Eagles fan. Hence the usage of some.

Negation

- “No students are math majors.”
- To deny this statement, we need at least one instance in which a student does major in math.
- “Some students are math majors.”

Negation

- To summarize negation:
- All p are q is negated by Some p are not q
- No p are q is negated by Some p are q

CONJUNCTION p ^ q

- A conjunction is a compound statement that consists of 2 or more statements connected by the word and.
- And is represented by the symbol ^.
- p ^ q represents “p and q”.
- Example:
p: Jerry Seinfeld is a comedian.

q: Jerry Seinfeld is a millionaire.

Express the following in symbolic form:

i. Jerry Seinfeld is a comedian and he is a millionaire.

ii. Jerry Seinfeld is a comedian and he is not a millionaire.

Conjunction

- Using the symbolic representations
p: The lyrics are controversial.

q: The performance is banned.

Express the following in symbolic form:

a. “The lyrics are controversial and the performance is banned.”

b. “The lyrics are not controversial and the performance is not banned.”

Answers:

a. p ^ q

b. ~p ^ ~q

DISJUNCTION p v q

- When you connect statements with the word or you form a disjunction.
- Or is represented by the symbol v.
- p v q is read as “p or q”.
- Using the p and q from the last slide, write out in words p v q, and p v ~q.
- p v q is “the lyrics are controversial or the performance is banned.”
- p v ~q is “the lyrics are controversial or the performance is not banned.”

CONDITIONAL p q

- A conditional is of the form “if p then q”. This is also known as an implication. p is the hypothesis (or premise), and q is the. conclusion.
- The representation of “if p then q” is p q.
- Again use the p and q from the previous 2 slides.
- “If the lyrics are not controversial, the performance is not banned.”
- ~p ~q

Sec. 1.2 #28

- Using the following symbolic representations
- p: I am innocent.
- q: I have an alibi.
- express the following in words.
- A. p ^ q
- Answer: “I am innocent and I have an alibi.”
- B. p q
- Answer: “If I am innocent, then I have an alibi.”
- C. ~q ~p
- Answer: “If I do not have an alibi, then I am not innocent.”
- D. q v ~p
- Answer: “I have an alibi or I am not innocent.”

Sec. 1.2 #30

- Using the symbolic representations
- p: I am innocent.
- q: I have an alibi.
- r: I go to jail.
- Express the following in words.
- A. (p v q) ~r
- B. (p ^ ~q) r
- C. (~p ^ q) v r
- D. (p ^ r) ~q

Sec. 1.2 #30

- If I am innocent or have an alibi, then I do not go to jail.
- If I am innocent and do not have an alibi, then I go to jail.
- I am not innocent and I have an alibi or I go to jail.
- If I am innocent and go to jail, then I do not have an alibi.

Sec. 1.2 #23

- Translate the sentence to symbolic form.
- If you drink and drive, you are fined or you go to jail.
- p: You drink.
- q: You drive.
- r: You are fined.
- s: You are jailed.
- Answer: (p ^ q) (r v s).

Sec. 1.2 #14

- Translate into symbolic form.
- “No whole number is greater than 3 and less than 4.”
- p: A whole number.
- q: A number greater than 3.
- r: A number less than 4.
- Answer: ~p (q ^ r)

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