LOGIC

l A conditional statement is a statement that can be expressed in "if...then..." form.

n Hypothesis: This is the part of the sentence that follows the word "If..."

n Conclusion: This is the part of the sentence that follows the word "then..."

It may be necessary when working with conditional statements to rewrite the sentence so that it is in "If...then..." form.

Example:

"All surfers like big waves."

"If you are a surfer, then you like big waves."

Putting a sentence in "If...then..." form before beginning your investigation of a conditional statement will make your work easier.

l The converse of a conditional statement is formed by interchanging the hypothesis and conclusion of the original statement, with the words "if" and "then" fixed.

Example:

Conditional:  "If the space shuttle was launched, then a cloud of smoke was seen."

Converse:  "If a cloud of smoke was seen, then the space shuttle was launched."

It is important to note that the converse of a true/false conditional statement is NOT necessarily true/false as the original statement.

l The inverse of a conditional statement is formed by negating the hypothesis and negating the conclusion of the original statement.

Example:

Conditional:  "If you grew up in Alaska, then you have seen snow."

Inverse:  "If you did not grow up in Alaska, then you have not seen snow."

You may also use other words (in addition to NOT) to create the negation.

It is important to note that the inverse of a true/false conditional statement is NOT necessarily true/false as the original statement.

l The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. In other words, it does BOTH the jobs of the INVERSE and the CONVERSE.

Example:

Conditional:  "If 9 is an odd number, then 9 is divisible by 2."

Contrapositive:  "If 9 is not divisible by 2, then 9 is not an odd number."

An important fact to note is that:

If the original statement is TRUE, the contrapositive is TRUE, and vice versa;

If the original statement is FALSE, the contrapositive is FALSE, and vice versa.

In this case, they are said to be logically equivalent.

Something important to be commented on:

l Sometimes there is a big premise in the statement which is neither hypothesis nor conclusion.

Example:

Conditional: "For any integer a, if a is divisible by 6, then a is even."

Contrapositive: " For any integer a, if a is odd, then a is not divisible by 6."

Conditional: "Empty set has no element in it."

Contrapositive: "If a set has some element in it, then it is not empty."

l Negation pairs:

n "and" versus "or"

Example:

Conditional: "For any real numbers a, b, if ab=0, then either a=0 or b=0."

Contrapositive: " For any real numbers a, b, if a is not 0 and b is not 0, then ab is not 0."

n "any", "all" versus "there is/exists a/some"

Example:

Statement: "All the students in the class are male."

Negation: "There is a student in the class, who is female."

Conditional: "If a subset A of the set of real numbers is bounded above, then there is a real number x, such that x is bigger than or equal to a, for all a in A."

Contrapositive: " A is a subset of the set of real numbers, if for any real number x, x<a for some a in A, then A is not bounded above."

Although using these negation pairs are helpful when you do your operations, such as negation, contrapositive, on a statement, there is no literal translation. Use your common sense to do them.