# 1. Propositions A proposition is a declarative sentence that is either true or false. Examples of propositions: The Moon is made of green cheese. Trenton.

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Propositions A proposition is a declarative sentence that is either true or false. Examples of propositions: The Moon is made of green cheese. Trenton is the capital of New Jersey. Toronto is the capital of Canada. 1 + 0 = 1 0 + 0 = 2 Examples that are not propositions. Sit down! What time is it? x + 1 = 2 x + y = z 2

Propositional Logic (or Calculus) Constructing Propositions Propositional Variables: p, q, r, s, … The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. Compound Propositions: constructed from other propositions using logical connectives Negation ¬ Conjunction ∧ Disjunction ∨ Implication → Biconditional ↔ 3

Compound Propositions: Negation The negation of a proposition p is denoted by ¬p and has this truth table: Example: If p denotes “The earth is round” then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” p¬p¬p TF FT 4

Conjunction The conjunction of propositions p and q is denoted by p ∧ q and has this truth table: Example: If p denotes “I am at home” and q denotes “It is raining” then p ∧q denotes “I am at home and it is raining.” pqp ∧ q TTT TFF FTF FFF 5

Disjunction The disjunction of propositions p and q is denoted by p ∨q and has this truth table: Example: If p denotes “I am at home” and q denotes “It is raining” then p ∨q denotes “I am at home or it is raining.” pqp ∨q TTT TFT FTT FFF 6

The Connective Or in English In English “or” has two distinct meanings. Inclusive Or: For p ∨q to be T, either p or q or both must be T Example: “CS 202 or Math 120 may be taken as a prerequisite.” Meaning: take either one or both Exclusive Or (Xor). In p ⊕q, either p or q but not both must be T Example: “Soup or salad comes with this entrée.” Meaning: do not expect to get both soup and salad pqp ⊕q TTF TFT FTT FFF 7

Implication If p and q are propositions, then p →q is a conditional statement or implication, which is read as “if p, then q ” p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). Example: If p denotes “It is raining” and q denotes “The streets are wet” then p →q denotes “If it is raining then streets are wet.” pqp →q TTT TFF FTT FFT 8

Understanding Implication In p →q there does not need to be any connection between p and q. The meaning of p →q depends only on the truth values of p and q. Examples of valid but counterintuitive implications: “If the moon is made of green cheese, then you have more money than Bill Gates” -- True “If Juan has a smartphone, then 2 + 3 = 6” -- False if Juan does have a smartphone, True if he does NOT 9

Understanding Implication View logical conditional as an obligation or contract: “If I am elected, then I will lower taxes” “If you get 100% on the final, then you will earn an A” If the politician is elected and does not lower taxes, then he or she has broken the campaign pledge. Similarly for the professor. This corresponds to the case where p is T and q is F. 10

Different Ways of Expressing p →q if p, then q if p, q p implies q q if p q when p q whenever p Note: (p only if q) ≡ (q if p) ≢ (q only if p) It is raining → streets are wet: T It is raining only if streets are wet: T Streets are wet if it is raining: T Streets are wet only if it’s raining: F 11 q follows from p q unless ¬p p only if q p is sufficient for q q is necessary for p

Sufficient versus Necessary Example 1: p →q : get 100% on final → earn A in class p is sufficient for q: getting 100% on final is sufficient for earning A q is necessary for p: earning A is necessary for getting 100% on final Counterintuitive in English: “Necessary” suggests a precondition Example is sequential Example 2: Nobel laureate → is intelligent Being intelligent is necessary for being Nobel laureate Being a Nobel laureate is sufficient for being intelligent (better if expressed as: implies) 12

Converse, Inverse, and Contrapositive From p →q we can form new conditional statements. q →p is the converse of p →q ¬ p → ¬ q is the inverse of p →q ¬q → ¬ p is the contrapositive of p →q How are these statements related to the original? How are they related to each other? Are any of them equivalent? 13

Converse, Inverse, and Contrapositive Example: it’s raining → streets are wet 1. converse: streets are wet → it’s raining 2. inverse: it’s not raining → streets are not wet 3. contrapositive: streets are not wet → it’s not raining only 3 is equivalent to original statement 1 and 2 are not equivalent to original statement: streets could be wet for other reasons 1 and 2 are equivalent to each other 14

Biconditional If p and q are propositions, then we can form the biconditional proposition p ↔q, read as “ p if and only if q ” Example: If p denotes “You can take a flight” and q denotes “You buy a ticket” then p ↔q denotes “You can take a flight if and only if you buy a ticket” True only if you do both or neither Doing only one or the other makes the proposition false pqp ↔q TTT TFF FTF FFT 15

Expressing the Biconditional Alternative ways to say “p if and only if q”: p is necessary and sufficient for q if p then q, and conversely p iff q 16

Compound Propositions conjunction, disjunction, negation, conditionals, and biconditionals can be combined into arbitrarily complex compound proposition Any proposition can become a term inside another proposition Propositions can be nested arbitrarily Example: p, q, r, t are simple propositions p ∨q, r ∧t, r →t are compound propositions using logical connectives (p ∨q) ∧ t and (p ∨q) →t are compound propositions formed by nesting 17

Precedence of Logical Operators OperatorPrecedence  1     2323     4545 18 With multiple operators we need to know the order To reduce number of parentheses use precedence rules Example: p ∨q → ¬ r is equivalent to (p ∨q) → ¬ r If the intended meaning is p ∨(q → ¬ r ) then parentheses must be used

Truth Tables of Compound Propositions Construction of a truth table: Rows One for every possible combination of values of all propositional variables Columns One for the compound proposition (usually at far right) One for each expression in the compound proposition as it is built up by nesting 19

Example Truth Table Construct a truth table for pqr p  q rr p  q →  r TTTTFF TTFTTT TFTTFF TFFTTT FTTTFF FTFTTT FFTFFT FFFFTT 20

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