# [MATH V1207] - Final Exam Guide - Everything you need to know! (30 pages long)

107 views30 pages
Published on 29 Nov 2016
School
Department
Professor
Columbia
MATH V1207
FINAL EXAM
STUDY GUIDE
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 30 pages and 3 million more documents.

Unlock document

This preview shows pages 1-3 of the document.
Unlock all 30 pages and 3 million more documents.

John Staunton Rudiments of Logic Honors Math A
I. Introduction to Statements
a. A statement is an assertion that is true or false but not both
i. For example: P = “The Earth is flat”
ii. Q = “Elvis is dead”
iii. The truth value of a statement P is T if it is true, and F if it is false
b. The negation of a statement () is equivalent to the statement “P is false.”
i. The truth value of the negation is the opposite of the original
ii. If P is false, the statement “P is false” is true
iii. Example: = “The Earth is not flat” (NOT “The Earth is round”)
iv. = “Elvis is not dead
c.
d. The conjunction of two statements P and Q, denoted as , is true if P and Q are both true
i. It would be false otherwise
ii. Example: = “The Earth is flat and Elvis is dead”
e. Proposition: is always false
i. Proof: If P is true, then ~P is false, so the conjunction is false since if it a T in conjunction with a
F, it is F by definition
f. The disjunction of two statements P and Q, denoted by is the statement that is false if both P and Q are
false and true otherwise
i. Proposition: The statement is always true
ii. Proof: If P and Q are both true, then the conjunction is true by definition, so the whole
disjunction is true. Otherwise, one of the P and Q is false so one of ~P or ~Q is true, so that
disjunction is true, and so is whole statement, which is a disjunction
g. The conditional statement “P implies Q” or “If P, then Q” denoted by P  Q is the statement
i. The only way this can be true is if Q is true
ii. P is the hypothesis and Q is the conclusion
iii. If P is false, then “P  Q” is said to be vacuously true
iv. Example: P is “We’re all in NYC” and Q is “We’re all in NYS”
v. To prove P  Q is true, it suffices to assume P is true and to deduce Q is true
h. Proposition is always true
i. If P in conjunction with Q is true, then both P is true and Q is true, which means that P is true
i. A bi-conditional statement is an “if and only if statement” – “P is true if and only if Q is true”
i. This is denoted by
ii. They have to both be true or both be false for the statement itself to be true
iii. This can also be written as
II. Types of Proofs and Proofs for Proofs
a. Proposition 1: Direct Proof
i. Supposed is true, and P is true, so Q is true
b. Proposition 2: Indirect Proof
i. Supposed are true, then ~P is true
ii. Since P Q is T, then is true and since ~Q is true, ~P must be true otherwise the second
statement is false
c. Proposition 3: Proof by Contradiction
i. Suppose the statement is true, then ~P is true
ii. Since the conditional true and the latter half is always false, P must be false and ~P must be true
d. Proposition 4: Proof by Cases
1
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 30 pages and 3 million more documents.

[Type text] [Type text] [Type text]
i. Suppose is true and , then R must be true.
2
find more resources at oneclass.com
find more resources at oneclass.com
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 30 pages and 3 million more documents.

## Document Summary

Honors math a: introduction to statements, a statement is an assertion that is true or false but not both i. ii. iii. For example: p = the earth is flat . The truth value of a statement p is t if it is true, and f if it is false: the negation of a statement () is equivalent to the statement p is false. i. ii. iii. iv. The truth value of the negation is the opposite of the original. If p is false, the statement p is false is true. Example: = the earth is not flat (not the earth is round ) = elvis is not dead c: the conjunction of two statements p and q, denoted as , is true if p and q are both true i. ii. Example: = the earth is flat and elvis is dead : proposition: is always false i.