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

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Published on 29 Nov 2016

School

Department

Course

Professor

Columbia

MATH V1207

FINAL EXAM

STUDY GUIDE

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

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i. Suppose is true and , then R must be true.

2

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## 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.