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Mathematical Statements and truth Values

Logic & types of statements

A mathematical sentence is a sentence that states a fact or contains a complete idea. A sentence that can be judged to be true or false is called a statement , or a closed sentence.


Import terms in Logic & Mathematical Statements

  • Negation

      Indicates the opposite, usually employing the word not. The symbol to indicate negation is : ~
      Original Statement Negation of statment
      Today is monday. Today is not monday.
      That was fun. That was not fun.


  • Conjunction

      In logic, a conjunction is a compound sentence formed by using the word and to join two simple sentences. The symbol for this is Λ. (whenever you see Λ read 'and') When two simple sentences, p and q, are joined in a conjunction statement, the conjunction is expressed symbollically as p Λ q.
      Simple Sentences Compound Sentence : conjunction
      p: Joe eats fries.
      q: Maria drinks soda.
      p Λ q : Joe eats fries, and maria drinks soda.


  • Disjunction

      In logic, a disjunction is a compound sentence formed by using the word or to join two simple sentences. The symbol for this is ν. (whenever you see ν read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbollically as p ν q.
      Pneumonic: the way to remember the symbol for disjuntion is that, this symbol ν looks like the 'r' in or, the keyword of disjunction statements.
      Simple Sentences Compound Sentence : disjunction
      p: The clock is slow.
      q: The time is correct.
      p ν q : The clock is slow, or the time is correct.


  • Warning and caveat: The only way for a disjunction to be a false statement is if BOTH halves are false. A disjunction is true if either statement is true or if both statements are true! In other words, the statement 'The clock is slow or the time is correct' is a false statement only if both parts are false! Likewise, the statement 'Mr. G teaches Math or Mr. G teaches Science' is true if Mr. G is teaches science classes as well as math classes!

    The Conditional



    In logic, a conditional statement is compound stentence that is usually expressed with the key words 'If....then...'. Using the variables p and q to represent two simple sentences, the conditional "If p then q" is expressed symbollically as p q.
    Simple Sentences Compound Sentence : Conditional
    p: You are absent
    q: You have a make up assignment to complete.
    p q : If you are absent, then you have a make up assignment to complete.
    Note: The word 'then' is optional, and a conditional will often not include 'then'. The example above could have been expressed: If you are absent, you have a make up assignment to complete.
    Truth values of Conditionals
    The only time that a conditional is a false statement is when a true "if" clause leads to a false "then" clause. For example, the conditional "If you are on time, then you are late." is false because when the "if" clause is true, the 'then' clause is false. THEREFORE, the entire statement is false.
    Example of a false conditional
    If Clause Then clause Entire Statment
    p q p q
    you are late you are on time If you are late, then you are on time.
    When p is true then q is false The entire statment is false
    True False False


    Warning and Caveat The opposite situation does not lead to a false statement. A false 'if' clause and a true 'then' clause creates a true statement. (Seems counterintuitive at first!) See the table below for an example.
    If Clause Then clause Entire Statment
    p q p q
    a human is a cat then squares have corners If a human is a cat, then squares have corners.
    When p is false q is true The entire statment is true.
    False True True
    Explanation: The if clause is always false (humans are not cats) , and the then clause is always true (squares always have corners). And the entire statement is true



Practice Problems

The practice problems below cover the truth values of conditionals, disjunction, conjunction, and negation.
Part I.
Let a represent "We go to school on Memorial Day."
Let b represent "Memorial Day is a holiday."
Let c represent "We work on Memorial Day."
Be prepared to express each statement symbolically, then state the truth value of each mathematical statement.
Practice problem 1)
Statement: We work on Memorial Day or Memorial Day is a holiday.
Answer
Practice problem 2)
Statement: Memorial Day is a holiday and we do not work on Memorial Day.
Answer
Practice problem 3)
Statement: If we go to school on Memorial Day, then we work on Memorial Day.
Answer
Practice problem 4)
Statement: We do not go to school on Memorial Day implies that we work on Memorial Day.
Answer
Practice problem 5)
Statement: We work on Memorial Day if and only if we go to school on Memorial Day.
Answer
Practice problem 6)
Statement: If we do not go to school on Memorial Day and Memorial day is a holiday, then we do not work on Memorial Day.
Answer