Logic operators

Logic is one of the most interesting branches of math. It is a very broad discipline of which we will learn only a small part here: matters of “true and false”.

Simple statements are facts that can be said to be either true or false. Note: questions, wishes or exclamations are not considered logical statements since they can’t be regarded as true or false (“Where is the school?” is not true nor false!).

An example of a true statement is: The sun shines during the day.
An example of a false statement is: The sun shines at night.

There are cases where it is impossible to determine if a statement is true or false:

• This can happen if some details are missing, for example: My friend went to the movies last night.
• The statement is semantically problematic, for example: I am lying. Is he telling the truth or lying?!

In addition to the concepts of true and false, logic also deals with negation. This concept may be explained by means of the following example: “It does not rain in November” is a statement that negates the statement “It rains in November”. The negation of a statement is false whenever the original statement is true, and vice versa.
This definition is also appropriate when we don’t know if the statement is true or false. If it is false, its negation is true, and vice versa.

The negation operation is an operator applied to a statement! There are other logic operators as well.

The operator and (conjunction): Suppose that a is one statement and b is another statement. Statement a and b is true when both a and b are true statements. In other words, when a statement is made up of two statements connected by the word and:

• When both statements are true, the whole statement is true.
• When at least one of the statements is false, the whole statement is false.

For example: If statement a is “There are clouds in the sky when it rains” (a true statement), and statement b is “Puddles are formed when it rains on the road” (a true statement), then the statement a and b: “There are clouds in the sky when it rains and puddles are formed when it rains on the road” is a true statement.

The operator or (disjunction): Suppose that a is one statement and b is another statement. Statement a or b is true when at least one of the two statements: a ,b, is true.

For example: If statement a is “There are clouds in the sky when it rains” (a true statement), and statement b is “A cat usually has three legs” (a false statement), then the statement a or b, “There are clouds in the sky when it rains or a cat usually has three legs” is a true statement.

Another example : Statement a: “Daniel went to school” (a true statement) Statement b: “Hannah went to school” (a true statement) “Daniel went to school or Hannah went to school” (a true statement).

Note that there is a semantic difference between “or” in the language and or in logic. In spoken language when we say “or” we usually mean that only one of the two options is true, and in logic it means that at least one of the two options is true.

Watch this video to see a slideshow summarising logic operators.