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.

Important terms in Logic & Mathematical Statements

Negation

Indicates the opposite, usually employing the word not.

The symbol to indicate negation is : ~

Original Statement Negation of Statement
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 the word and to join two simple sentences.

The symbol for this is $$ Λ $$. (whenever you see $$ Λ $$ , just read 'and') When two simple sentences, p and q, are joined in a conjunction statement, the conjunction is expressed symbolically 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 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 symbolically as p $$ ν$$ q.
Pneumonic: the way to remember the symbol for disjunction 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 sentence 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 symbolically as p $$\rightarrow$$ q arrow right

Simple Sentences Compound Sentence: Conditional
p: You are absent
q : You have a make up assignment to complete.
p $$ \rightarrow $$ 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 omit the word '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 the if clause is true and the then clause is false .

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 Statement
p q p arrow right 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 statement 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 Statement
p q p arrow right 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 statement 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.

Problem 1

Statement: We work on Memorial Day or Memorial Day is a holiday.

Statement in symbols Truth value of parts Truth value of entire statement
c ν b F ν T True statement
Problem 2

Statement: Memorial Day is a holiday and we do not work on Memorial Day.

Statement in symbols Truth value of parts Truth value of entire statement
b Λ ~c T Λ ~F = T Λ T True statement
Problem 3

Statement: If we go to school on Memorial Day, then we work on Memorial Day.

Statement in symbols Truth value of parts Truth value of entire statement
a arrow right c F arrow right F True statement
Explanation: this is a conditional statement, and the 'if' clause is false because we do not go to school on memorial. Also, the 'then' clause is false because we do notwork on Memorial Day. However, this statement is a true statement in its entirety. Remember the only way that a conditional is a false statement is when a true 'if' clause leads to a false 'then' clause (ie when T arrow right F) (example of false conditional)
Problem 4

Statement: We do not go to school on Memorial Day implies that we work on Memorial Day.

Statement in symbols Truth value of parts Truth value of entire statement
~a arrow right c ~F arrow right F = T arrow right F False statement
Problem 5

Statement: We work on Memorial Day if and only if we go to school on Memorial Day.

Statement in symbols Truth value of parts Truth value of entire statement
c arrow both ways a F arrow both ways F True statement
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.

Statement in symbols Truth value of parts Truth value of entire statement
(~a Λ b) arrow right ~c (~F Λ T) arrow right ~ F
(T Λ T) arrow right ~F
T arrow right ~F
T arrow right T
True statement