Remember the following.

• true is always true,
• true can never be false,
• false is always false, and
• false can never be true.

Here are two simple operators.

• "Eqv" means "is equal to" (equivalence)
• "Neqv" means "is not equal to" (nonequivalence)

These operators are relational operators as they relate quantities, as opposed to being logical operators (and, or, not, imply, etc.).

The following are true expressions.

• true
• true Eqv true
• true Neqv false
• false Eqv false
• false Neqv true

The following are false expressions.

• false
• true Neqv true
• true Eqv false
• false Neqv false
• false Eqv true

There are three primary logical operations.

• Logical negation (i.e., Not, "Not", etc.) means not.
• Logical conjunction (i.e., And, "And", etc.) means both.
• Logical disjunction (i.e., Or, "Or", etc.) means one or the other or both.

Some other common logical operations include the following.

• Logical equivalence or equals (i.e., Eqv, "Eqv", etc.) means both operands are the same.
• Logical nonequivalence or exclusive or (i.e., Xor, "Xor", etc.) means both operands are different.

Logical negation returns the opposite of the operand.

• Not True is False
• Not False is True

Some other symbols for Not:

• ¬ (math notation)
• ! (C-based languages)
• not (Python, Lua, Pascal, etc.)

Logical negation (i.e., "Not") is related to the complement operation of sets.

Here is a mathematical definition of complement.


Question: What is the opposite of not in?

The opposite of not in is in

• "in" is the opposite of "out".
• "out" is the meaning of "not in".
• So, "in" is the opposite of "not in".

See how easy it is to get mixed up, if you are not careful.

Is the following sentence true or false?

This sentence is false.

This is a self-referential statement since it refers to itself.

Self-referential statements can cause logical problems (i.e., paradoxes).

Is "no" your answer to this question? Answer either "yes" or "no".

The conjunction of two planets happens when the two planets are both inline with the sun.

Example: Conjunction of Venus and the Earth.

A logical conjuction is true of both operands are true.

• False And False is False
• False And True is False
• True And False is False
• True And True is True

Some other symbols for And:

• ∧ (math notation), ∩ (set notation, intersection)
• && (C-based languages , & is the bitwise conjunction)
• and (Python, Lua, Pascal, etc.)

Logical conjunction is the same as the natural language "and".

The natural language "and" is the same as the logical "and".

Example: To graduate, you must take and pass both CS201 and CS340 (and some others, ignored here)

That is, you must take both courses.

Logical conjunction (i.e., "And") is related to the intersection operation of sets.

Here is a mathematical definition of set intersection.

Logical disjunction is true if either or both operands are true.

• False Or False is False
• False Or True is True
• True Or False is True
• True Or True is True

Some other symbols for Or:

• ∨ (math notation), ∪ (set notation, union)
• || (C-based languages , | is the bitwise conjunction)
• or (Python, Lua, Pascal, etc.)

Logical disjunction (i.e., "Or") is related to the intersection operation of sets.

Here is a mathematical definition of set union

Logical disjunction is not the same as the natural language "or", which is an exclusive "or" and not an inclusive "or".

Question: Would you like a piece of pie or a piece of cake?

A logical answer is, "Yes", That is, I would like either a piece of pie, or a piece of cake, or both. I'll take both.

The natural language answer is to pick one of the two, but not both.

As a consequence, you must always be careful when using the word "or".

• The logical "or" is logical disjunction.
• The natural language "or" is an exclusive "or", or nonequivalence.

The query command is something like "find all pages that contain the specified text".

Search engines on the Internet allow logical operations to be used to limit the search criteria.

Question: When might we need more than two values for logical questions?

In a modal dialog box, the user may need to decide between yes, no, and some other choices.


A program that requests a user response to a question. Very often, the valid responses are yes, no, and escape (gracefully avoid the question).

Determining the sign of an integer, which can be positive, negative, or zero (neither positive nor negative).

How many choices were there for Paul Revere's famous ride?

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