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• Proposition


  A declarative sentence that assigns one and only one of the two possible truth values: true (1) or false (0)
• Logical methods are used in mathematics to prove theorems, in computer science to verify the correctness of programs, in the natural and physical sciences to draw conclusions from experiments, in the social sciences and in our everyday lives to solve a multitude of problems
• Logical Connectives

  • Negation
  • Conjunction
  • Disjunction
  • Inclusive Disjunction
  • Exclusive Disjunction
• Truth Value


  The assigned value to a given proposition
• Truth Table


  A table which summarizes the truth values of propositions, displaying all the possible combinations of the given proposition
• How to Construct a Truth Table

  1. Step 1: Prepare all possible combinations of truth values for propositional variables

  2. Step 2: Obtain the truth values of each connective and put these truth values in a new column
• Negation


  The statement "It's not the case that p" denoted as ¬p or ~p, read as "not p"
• Inserting the word "not" in a statement is a way to express negation
• Other symbols like "-" and ">" can also be used as negation
• Negation


  • It is false that p
  • It is not the case that p
• Negation


  • ~(Today is Friday)
  • ~(The sun is not shining)
  • ~(Five is an even number)
  • ~(I like Math)
• Conjunction


  The propositions "p and q" denoted as p ∧ q, read as "the conjunction of p and q"
• "but" has the same logical meaning as "and"
• Other words used to conjoin two propositions: moreover, furthermore, yet, still, however, also, nevertheless, although, and so forth, comma (,) and colon (:)
• Conjunction


  • (Today is Friday) ∧ (It is raining today)
  • (I am sick) ∧ (I cannot take the exam)
• Disjunction


  The "disjunction of p and q" denoted as p ∨ q, read as "p or q"
• Inclusive OR


  If p and q are propositions, p OR q is true if either p is true or q is true or if both p and q are true
• Exclusive OR


  If p and q are propositions, the "exclusive or" of p and q is denoted as p ⊕ q, meaning that strictly one of the propositions must be true in order for the exclusive disjunction to be true
• "unless" may also be used in expressing the disjunction of two propositions
• Inclusive Disjunction


  • (Plaridel is the capital of Bulacan) ∨ (Malolos is one of the cities found in Region III)
  • (3 is an even number) ∨ (A century is 100 years)
• Exclusive Disjunction


  • (I am looking at my seatmate) ⊕ (I am looking at my teacher)
  • (I can take a plane going to Romblon) ⊕ (I can take a ferry going to Romblon)
• Disjunction


  Disjunction of two propositions
• Disjunction: Truth Tables
• Inclusive Disjunction


  p v q
• Inclusive Disjunction Truth Table


  • 1 1 1
  • 1 0 1
  • 0 1 1
  • 0 0 0
• Exclusive Disjunction


  p ⊕ q
• Exclusive Disjunction Truth Table

  • 1 1 0
  • 1 0 1
  • 0 1 1
  • 0 0 0
• Inclusive Disjunction Examples


  • Plaridel is the capital of Bulacan or Malolos is one of the cities found in Region III
  • 3 is an even number or a century is 100 years
• Exclusive Disjunction Examples


  • I am looking at my seatmate or I am looking at my teacher
  • I can take a plane or a ferry going to Romblon
• Note on Compound Propositions
• Conditional Propositions


  If p and q are propositions, the compound statement "if p, then q" is called an implication or conditional statement and is denoted by p → q. p is the hypothesis (or antecedent), q is the conclusion (or consequent).
• Conditional Propositions Examples


  • If I am late then I cannot take the seatwork
  • If today is Monday then I have a test today
• Conditional Propositions Examples 2


  • If it is not a long weekend, then Lucky is not going to watch Riverdale
  • If it is a long weekend, then I will stay at home
  • I will stay at home if it is a long weekend
  • I will stay at home whenever it is a long weekend
• Note on Conditional Proposition
• Conditional Proposition Truth Table


  • 1 1 1
  • 1 0 0
  • 0 1 1
  • 0 0 1
• Note on Conditional Proposition


  Converse: q → p, Contrapositive: ~q → ~p, Inverse: ~p → ~q
• Note on Conditional Proposition Examples

  • Implication: If it is hot, then I will go to the mall

  Converse: If I will go to the mall, then it is hot

  Contrapositive: If I will not go to the mall, then it is not hot

  Inverse: If it is not hot, then I will not go the mall
• Conditional Proposition Truth Table


  • 1 1 0 0 1 1 1 1

  1 0 0 1 0 1 1 0

  0 1 1 0 1 0 0 1

  0 0 1 1 1 1 1 1
• Biconditional Propositions


  If p and q are propositions, the compound proposition "p if and only if q" is called a biconditional proposition and is written p ↔ q.
• Biconditional Propositions Examples


  • David is the son of Ricky if and only if Ricky is the father of David

  12 is divisible by 2 if and only if 12 is even