Mathematical Statements and truth Values

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.




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negation | conjunction | disjunction | conditional | practice #1

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• 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.




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negation | conjunction | disjunction | conditional | practice #1

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• 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!


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negation | conjunction | disjunction | conditional | practice #1

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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

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.”