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OPTIMIZED LOGIC STUDY GUIDE (CH 2.2-3.4)
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1
Conditional (p→q)
False only when p is true and q is false.
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2
Truth table for p→q
Only false in the row where p is true and q is false.
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3
Vacuously true
Occurs when the hypothesis p is false.
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4
Key equivalence of p→q
p→q is equivalent to ~p∨q.
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5
Negation of p→q
~(p→q) is equivalent to p∧~q.
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6
Contrapositive
~q→~p, which is equivalent to the original conditional.
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7
Converse
q→p, which is not equivalent to the original conditional.
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8
Inverse
~p→~q, which is not equivalent to the original conditional.
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9
Meaning of 'p only if q'
Represents the conditional p→q.
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10
Biconditional (p↔q)
Defined as (p→q)∧(q→p).
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11
Necessary condition
Saying 'You can't have p without q' means p→q.
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12
Sufficient condition
Saying 'q automatically gives you p' means q→p.
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13
Modus Ponens
If p→q and p are true, then q must be true.
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14
Modus Tollens
If p→q and ~q are true, then ~p must be true.
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15
Transitivity
If p→q and q→r are true, then p→r must be true.
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16
Elimination
If p∨q and ~p are true, then q must be true.
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17
Division into Cases
If p∨q, p→r, and q→r are true, then r must be true.
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18
Contradiction Rule
If ~p leads to a contradiction, then p must be true.
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19
Converse Error
The fallacy where p→q and q leads to the invalid conclusion of p.
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20
Inverse Error
The fallacy where p→q and ~p leads to the invalid conclusion of ~q.
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21
Predicate P(x)
A statement that contains variables.
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22
Truth set
The set of values that make the predicate P(x) true.
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23
Universal quantifier (∀x)
Indicates 'for all x'.
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24
Existential quantifier (∃x)
Indicates 'there exists at least one x'.
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25
Universal statement form
∀x, P(x) or ∀x, if P(x) then Q(x).
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26
Negation of universal statements
~(∀x, P(x)) is equivalent to ∃x, ~P(x).
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27
Negation of existential statements
~(∃x, P(x)) is equivalent to ∀x, ~P(x).
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28
Negating Universal Conditional
~(∀x, P(x)→Q(x)) is equivalent to ∃x, P(x)∧~Q(x).
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29
Order of quantifiers
The order matters: ∀x, ∃y, P(x,y) is not the same as ∃y, ∀x, P(x,y).
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30
Negating multiple quantifiers
~(∀x, ∃y, P(x,y)) is equivalent to ∃x, ∀y, ~P(x,y).
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31
Universal Instantiation
If ∀x, P(x) is true, then P(a) is true for any specific a.
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32
Universal Modus Ponens
If ∀x, P(x)→Q(x) and P(a) are true, then Q(a) must be true.
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33
Universal Modus Tollens
If ∀x, P(x)→Q(x) and ~Q(a), then ~P(a) must be true.
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34
Venn diagram for 'All A are B'
Represented as A circle inside the B circle.
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35
Venn diagram for 'No A are B'
Represented as non-overlapping circles.
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36
'→' Symbol
Represents a conditional statement (if-then).
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37
'↔' Symbol
Represents a biconditional statement (if and only if).
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38
'∧' Symbol
Represents conjunction (and).
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39
'∨' Symbol
Represents disjunction (or).
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40
'~' Symbol
Represents negation (not).
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41
'∀' Symbol
Represents the universal quantifier.
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42
'∃' Symbol
Represents the existential quantifier.
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43
'∈' Symbol
Represents 'element of'.
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44
'∴' Symbol
Represents 'therefore'.
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45
Operation order for logical expressions
Negation (~), followed by conjunction (∧) and disjunction (∨), then conditional (→) and biconditional (↔).
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