This section we will study symbolic logic which was developed in the late 17th century. This section we will study symbolic logic which was developed in the late 17th century
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.
Compound statements are known as negations, conjunctions, disjunctions, conditionals or combinations of each.
The negation of a statement is the denial of that statement. The symbolic representation is a tilde ~.
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.
“All of Mr. D’s students are Philadelphia Eagles fans.”
“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.
“No students are math majors.”
“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.”
To summarize 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
A conjunction is a compound statement that consists of 2 or more statements connected by the word and.
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.
Using the symbolic representations
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
When you connect statements with the word oryou form a disjunction.
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.”
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.
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.”
