MA.912.LT.4.1

• The answer to 2 plus 2 is 7. 
• The capital of Florida is Tallahassee. 
• The floor is lava. 
• Sentences that express opinions, questions, commands or whose truth value cannot be determined are not statements within the study of logic. 

• Are we there yet? (question) 
• The sky is beautiful. (opinion) 
• Take out the trash. (command) 
• This sentence is false. (truth value cannot be determined) 
• Propositional statements are designated with lower-case variables. The most commonly used variables in logic are p, q and r. 
  • The structure of the statement is defined as: 
    • p: The moon is made of cheese. 
    • q: All cows have feathers. 
• Propositional statements may be simple, quantified or compound. 
  • A simple propositional statement is a declarative statement that does not contain a connective. 
    • p: The TV is on. 
  • A quantified propositional statement contains a quantifier such as, all, none, or some. 
    • p: None of the remotes work. 
    • q: Some of the grass is green. 
  • A compound propositional statement includes multiple simple and quantified statements linked with logical connectives, such as ‘and,’ ‘or,’ “if…then” and “if and only if.” 
    • p: The car is broken and I have to go to work. 
    • q: If the sun is out, then it is raining. 
• Instruction includes the representation of logical connectives using logical symbolism and the use of proper terminology. Logical connectives are used to modify statements to produce new forms of statements, such as a negation, conjunction, disjunction, conditional, and biconditional.

• Logical modifiers and connectives will look like the example below: 
  • p: The light is on. 
  • q: The door is locked. 

• Conditional statements (p → q)contain a hypothesis (p) and a conclusion (q). In a conditional statement, p is the antecedent and q is the consequent. 
• Negating a sentence changes its truth value; a true statement becomes false, and a false statement becomes true. 
• Negating quantified and compound statements will require additional rules of logic. 
  • Negating quantified statements 

• p: All birds are blue. 
• ~p: Some birds are not blue. 
• Negating conjunction and disjunction statements use DeMorgan’s Laws (MA.912.LT.5.6): 

• p: The car is broken. 
• q: I have to go to work. 
• p ∧ q: The car is broken and I have to go to work. 
• ~(p ∧ q): It is not true that, the car is broken and I have to go to work. 
• ~p ∨ ~q: The car is not broken or I don’t have to go to work. 
• Negating conditional statements can be thought of as breaking a promise. 

• p: The car is broken. 
• q: I have to go to work. 
• p → q: If the car is broken, then I have to go to work. 
• ~(p → q): If it is not true that, if the car is broken, then I have to go work. 
• p ∧ ~q: The car is broken and I don’t have to go to work. 
• For mastery of this benchmark, students should be given opportunities to translate a variety of propositional statements into logical symbolism and vice versa while engaging in discussions that reflect on mathematical thinking of self and others (MTR.4.1). 
• Instruction includes connections to set operations (MA.912.LT.5.4) and Venn Diagrams (MA.912.LT.5.5).

Common Misconceptions or Errors

• Students may improperly negate quantified statements. 
  • For example, when asked to negate “All giraffes are orange,” students will respond with “No giraffes are orange.”
    To help address this, remind students that to negate a quantified statement of the type “All A are B,” we need to produce a counter-example of at least one A that is not B, or the shorter version “Some A are not B.” On the other hand, to negate “No A are B,” we need to find an example of an A that is also B, or “Some A are B.” 

• Students may improperly translate conditional statements where the hypothesis follows the conclusion statement. 
  • For example, let p: “The grass is green,” let q: “It is spring.” Students may incorrectly translate “the grass is green if it is spring” as p→q. 
    To help address this, remind students that p→q translates as “if p, then q” or “p implies q.” However, in the conditional statement provided, q is the hypothesis for the conclusion p. So the symbolic translation should be q→p. 

• Students may struggle when negating compound statements, such as conjunction, disjunction, and conditional.

Instructional Tasks

• For the simple statements:
  
  p: It is a frog. 
  q: It has feathers. 

• Part A. For each compound statement, write the symbolic form. 

• Part B. Justify your answers in writing by explaining the process you used to translate each compound statement. Compare your reasoning with others in your group.