9.3 KiB
title | TARGET DECK | FILE TAGS | tags | ||
---|---|---|---|---|---|
Propositional Logic | Obsidian::STEM | logic::propositional |
|
Overview
A branch of logic derived from negation (\neg
), conjunction (\land
), disjunction (\lor
), implication (\Rightarrow
), and biconditionals (\Leftrightarrow
). There exists a hierarchy of terms used to describe a string of English:
- A sentence is any grammatical string of words.
- A predicate is a sentence with free variables.
- A statement is a sentence that can be assigned a truth or false value.
- A predicate with free variables "plugged in" is a statement.
%%ANKI
Basic
What are the basic propositional logical operators?
Back: \neg
, \land
, \lor
, \Rightarrow
, and \Leftrightarrow
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic What is a propositional statement? Back: A declarative sentence which is either true or false. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What two categories do propositional statements fall within? Back: Atomic and molecular statements. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What is an atomic statement? Back: It cannot be broken up into smaller statements. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What is a molecular statement? Back: It can be broken up into smaller statements. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze A {molecular} statement can be broken up into {atomic} statements. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What distinguishes a sentence from a statement? Back: The latter is a sentence that can be derived a truth value. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What distinguishes a sentence from a predicate? Back: The latter is a sentence that contains free variables. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What distinguishes a predicate from a statement? Back: A statement does not contain free variables. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How are statements defined in terms of predicates?
Back: A statement is a predicate with 0
free variables.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Why is "3 + x = 12
" not a statement?
Back: Because x
is a variable.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Implication
Implication is denoted as \Rightarrow
. It has truth table
p |
q |
p \Rightarrow q |
---|---|---|
T |
T |
T |
T |
F |
F |
F |
T |
T |
F |
F |
T |
Implication has a few "equivalent" English expressions that are commonly used.
Given propositions P
and Q
, we have the following equivalences:
P
ifQ
P
only ifQ
P
is necessary forQ
P
is sufficient forQ
%%ANKI
Basic
What name is given to operand a
in a \Rightarrow b
?
Back: The antecedent
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What name is given to operand b
in a \Rightarrow b
?
Back: The consequent
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How does "P
if Q
" translate with \Rightarrow
?
Back: Q \Rightarrow P
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How does "P
only if Q
" translate with \Rightarrow
?
Back: P \Rightarrow Q
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How does "P
is necessary for Q
" translate with \Rightarrow
?
Back: Q \Rightarrow P
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
How does "P
is sufficient for Q
" translate with \Rightarrow
?
Back: P \Rightarrow Q
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Which of if or only if map to necessary? Back: if Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Which of if or only if map to sufficient? Back: only if Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Which logical operator maps to "if and only if"?
Back: \Leftrightarrow
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Which logical operator maps to "necessary and sufficient"?
Back: \Leftrightarrow
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the converse of P \Rightarrow Q
?
Back: Q \Rightarrow P
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic When is implication equivalent to its converse? Back: It's indeterminate. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What is the contrapositive of P \Rightarrow Q
?
Back: \neg Q \Rightarrow \neg P
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic When is implication equivalent to its contrapositive? Back: They are always equivalent. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
References
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.