Statements, premises, conclusions and conjectures
Statement
A statement is a sentence which is either true or false.
The following sentences are examples of statements:
- If x=2, then x-3=17.
- Every natural number is a real number.
- Every real number is negative.
Conditional statement
A conditional statement is an if-then statement.
The following statements are examples of conditional statements:
- If x=7, then x^2=49.
- If x=7, then x^2=50.
Premise
The premise of a conditional statement is the part of the conditional statement which is before the word "then".
Consider the conditional statement: "If x=3, then x-7=-4." The premise of the conditional statement is " x=3."
Conclusion
The conclusion of a conditional statement is the part of the statement which follows the word "then".
Consider the conditional statement: "If a is a real number, then a^2 is a natural number." The conclusion of the conditional statement is " a^2 is a natural number."
Conjecture
A conjecture is a statement that has not yet been rigorously proved. It is a guess based on data or intuition.
The next number in the sequence 2,4,6,8,10,12 can be conjectured to be 14 since each successive number is incremented by two.
Counterexamples
Counterexample
A counterexample to a statement is an example which illustrates that the statement is false.
Consider the following mathematical statement:
"Every natural number is even."
The number 3 is a natural number and is not even. Therefore, 3 is a counterexample to the above statement.
Consider the following mathematical statement:
"The sum of two prime numbers is even."
A counterexample to the above statement would be the numbers 2 and 3. This is because the numbers 2 and 3 are prime, but their sum 2+3=5 is not even.
To show a conditional statement is true, it must be proven true under the assumption of the premise. However, to show a conditional statement is false, only one counterexample is needed.
Conditionals and biconditions
Truth Tables
Truth Table
A truth table is a table which labels propositions as either true or false based on the truth values of other statements.
Suppose A and B are statements that can be either true or false. The statement " A and B " is true if and only if both A and B are true.
We can create a truth table to understand when the statement " A and B " is actually true. The table below contains three columns:
- A column for the statement " A ".
- A column for the statement " B ".
- A column for the statement " A and B ".
In the columns below A and B, we write all possible combinations of the statements being true or false. In the column with the statement " A and B ", we write whether the statement is true or false based off the truth values of the statements A and B. We use the letter T for true and the letter F for false.
Observe that the only time true appears under " A and B " is when both A and B are true.
Suppose A and B are statements which can be either true or false. The statement " A or B " is true if and only if at least one of A or B is true.
Below is the truth table for the statement " A or B ".
Observe that the only time false appears under " A or B " is when both A and B are false.
Conditional and biconditional statements
Conditional Statement
A conditional statement is an if-then statement. If A represents the premise of the conditional statement and B represents the conclusion of the conditional statement, then the sentence "If A is true, then B is true" is represented mathematically as:
A\Rightarrow B
The following is a conditional statement:
"If x=7, then 6x=42 "
The premise of the conditional statement is A= " x=7 ", and the conclusion of the statement is B= " 6x=42."
The following is a conditional statement:
"If n is a natural number, then n\left(n+1\right)\left(n+2\right) is divisible by 3."
The premise of the conditional statement is A= " n is a natural number", and the conclusion of the conditional statement is B= " n\left(n+1\right)\left(n+2\right) is divisible by 3."
Biconditional
Let A and B be two statements. A biconditional statement is a statement that combines two conditional statements into one. It can be expressed as "A if and only if B" and can be written as:
A\Leftrightarrow B
The following is a biconditional statement:
"An integer n is even if and only if n+1 is odd."
Converses, Inverses and Contrapositives
Converse of a conditional
The converse switches the hypothesis and conclusion of a conditional statement. If a conditional statement states A \Rightarrow B, then the converse is:
B\Rightarrow A
Consider the following conditional statement:
"If n is even, then 2n is even."
The converse of the conditional statement is:
"If 2n is even, then n is even."
Negation of a statement
Let A be a statement. The negation of A is the opposite of the statement and is written \sim A.
Consider the following statement:
" 17 is a prime number."
The negation of the statement is:
" 17 is not a prime number."
Inverse of a conditional
The inverse negates both the hypothesis and conclusion of a conditional statement. If a conditional statement states A \Rightarrow B, then the inverse is:
\sim A\Rightarrow \sim B
Consider the following conditional statement:
"If f\left(x\right) is a polynomial function, then the domain of f\left(x\right) is all real numbers."
The inverse of the above conditional statement is:
"If f\left(x\right) is not a polynomial function, then the domain of f\left(x\right) is not all real numbers."
Contrapositive of a conditional
The contrapositive of a conditional switches the hypothesis and conclusion of a conditional statement and negates both. If a conditional statement states A \Rightarrow B, then the contrapositive is:
\sim B\Rightarrow \sim A
Consider the following conditional statement:
"If the range of a function f\left(x\right) is \left(0{,}3\right], then f\left(1\right)=2."
The contrapositive to the conditional statement is:
"If f\left(x\right) is a function and f\left(1\right)\not=2, then the range of f\left(x\right) is not \left(0{,}3\right]."
The contrapositive of A\Rightarrow B is logically equivalent to the conditional A \Rightarrow B. Therefore, we can use the contrapositive as a different approach to prove a conditional.
Consider the following conditional statement:
"Suppose that a,b,n are natural numbers. If n does not divide ab, then n does not divide a or b."
The contrapositive to the statement is:
"If n divides a or b then n divides ab."
The contrapositive is easily seen to be true because if either a or b is a multiple of n, then so is ab. Therefore, the original conditional statement is true as well.
If a conditional statement is true, then it may not be the case that the converse or inverse conditional statement is true.
Consider the following conditional statement:
If n is an even number, then 2n is an even number.
The above conditional statement is clearly true because 2n is always an even number, regardless if n is even or not. However, consider the converse and inverse conditional statements:
- If 2n is even, then n is even.
- If n is not even, then 2n is not even.
Both statements are false and the number 3 is a counterexample to both statements. This is because:
- 2\left(3\right)=6 is even, but 3 is not even.
- 3 is not even, but 2\left(3\right) is even.