CSC 144 def

• ℤ =All integers

  ℤ⁺, ℕ⁺ = All positive integers

  ℤ* ℕ₀ = The non-negative integers

  ℤ_even = Even integers

  ℤ_odd = Odd integers

  ℚ = Rational numbers

  ℚ| = Irrational Numbers

  ℝ = The real values
• Commutativity

  Addition where 2+3=3+2 but 2-3 != 3-2
• Associativity

  Multiplication $(2*3)4=$$2(3*4)$
• Distributivity

  $a Δ (b ∨ c) =$$(a Δ b) ∨ (a Δ c) | (b ∧ c) Δ a =$$(b Δ a) ∧ (c Δ a)| 1214=$$12(10*4)$
• Transitivity a>b and b>c then a>c
• Rational Number is a ratio such as 1/2=0.5
• Irrational Number is a repeating number such as 1/3=0.3333....
• Union ( ∪ ): A ∪ B contains all elements of both set A and set B.
• Intersection ( ∩ ): C ∩ D contains only the elements present in both sets C and D.
• Difference ( − ): E − F contains only the elements of set E that are not also in set F.
• Complement ( \overline{G} ): Given a set G, \overline{G} = U − G, the set of available items, where U is the universe. Think all of the set minus the added set

  
  

  example say set A={1,2,3,4,5} set B={1,3,5}

  the answer is 2 and 3 since we are taking everything in A minus B
• Summation

  
• Product notation

  
• Modulo (%) the reminder of the divide number such as 7%3=1 and 7%11=7
• Integer Division (\) the quotient of the number such as 7\3=2 and 7\11=0
• Congruent modulo (m) is two numbers leave the same remainder when divided by (m)
• Philosophical Logic is the study of valid reasoning. Includes arguments and proofs.

  PS: think of logical words such as the sky is blue
• Mathematical Logic is the The use of formal English to represent the syntax such as if p then q (p → q)
• Well-Formed Formulae: A formula that is well-formed is one that is free of errors.
• Proposition is a claim that is either True or false with respect to a associated context
• logical operators words used to connect a logic such as "AND" , "OR" , "NOT" , "BUT"
• simple Proposition is a Proposition that contains no logical operators
• compound Propositions are propositions that are composed of two or more propositions.
• conjunctions are compound Propositions (∧) connected with "and" or "but" true when both are true think of "I got gas" and "I got a haircut""I got gas (and or but I) got a hair cut"
• Disjunctions are compound Propositionsconnects with "or" can be inclusive (V) or exclusive (⊕)

  (note: when exclusive, think one or the other but not both) you can get either 2 or 3 points, see how it can't be both 2 and 3?
• negation: it's (~) or "¬" placed before ℤ, line over it or a '.it reverses everything including signs and symbolsℤ: this is good¬ℤ: this is bad

  
• Logically Equivalent (≡) Two propositions yield the same result for any given input or truth values such as If x=3 and y=3, then x≡y.

  
  

  Propositions x and y are logically equivalent (x≡y) if the biconditional (↔) connecting them is a tautology, meaning it always evaluates to true. such as (x↔y) is a tautology.
• Tautology is a Proposition that is always true
• Contradiction is a Proposition that is always false
• Contingency is a Proposition that is neither a Tautology or Contradiction
• Conditional Proposition: A proposition expression in the form of "p->q", (if p, then q) p is the antecedent, hypothesis, sufficient q is the consequent, conclusion, necessary
• Vacuously true: a conditional proposition whose antecedent is false is Vacuously true
• Inverse: opposite
• converse: the converse of p → q is q → p
• "∈" represents the "element of" relationship in set theory.

  The expression "a ∈ B" is read as "a is an element of set B" or "a belongs to set B."
• "divides" or "is divisible by" (|) means no reminder

  4∣12 means "4 divides 12," and it is true

  7∣21 means "7 divides 21," and it is true
• When doing logical bit operations: just know 1 is true and 0 is false then in default Linux file permissions you have rwx and - where if you get 111 then the permission is rwx and if it's 000 then it's --- every slot replaces a vlaue
• Biconditional Proposition (↔)

  A logical connective representing a two-way relationship between two statements, true if both statements have the same truth value.

  • Example: P↔Q≡(P∧Q)∨(¬P∧¬Q)
  • Read as: "p if and only if q"
• De Morgan's Laws

  • First Law (Negation of Conjunction):
  • Symbol: ¬(p∧q)
  • Equivalent Expression: (¬p)∨(¬q)
  • Read as: "Not (p and q) is equivalent to (not p) or (not q)."
  • Second Law (Negation of Disjunction):
  • Symbol: ¬(p∨q)
  • Equivalent Expression: (¬p)∧(¬q)
  • Read as: "Not (p or q) is equivalent to (not p) and (not q)."
• Logical Equivalences

  with the power of math you can change the Logical Equivalences of things to other things

  • Example:
  • p∧(q∨r)≡(p∧q)∨(p∧r) (Distribution Law)
  • ¬(p∧q)≡¬p∨¬q (De Morgan's Law)