Mathematical Logic

Set Theory

• A set is a list of items, called elements, enclosed in curly braces and separated by commas.
• The empty set is a set containing no elements, and is denoted by empty curly braces, “{} “.
• A subset of another set only contains some or all of the other set.
• An intersection of two sets contains only the common elements found in both sets. It is also called a conjunction, and is denoted by a “ ∩ “.
  Example: If set A = {1, 2, 3} and set B = {3, 4, 5},
  then the intersection of set A and Set B is the set {3}:
  A ∩ B = {3}
• A union of two sets contains all the elements found in both sets. It is also called a disjunction, and is denoted by a “ U “.
  Example: If set A = {1, 2, 3} and set B = {3, 4, 5},
  then the union of set A and Set B is the set {1, 2, 3, 4, 5}:
  A U B = {1, 2, 3, 4, 5}
• A Venn Diagram is a graphical representation of sets using circles (potentially overlapping) to visually show set interactions.
  Example:
  
  

Boolean Logic

• Similar to Set Theory, but uses only two elements: True and False, T and F, 1 and 0.
• A Truth Table shows all possible combinations of inputs and the result of those combinations.
• The AND function takes two or more inputs and gives the result of True only when all the inputs are True. It is denoted by a “ ∙ “.
  Truth Table:

A	B	A ∙ B
F	F	F
F	T	F
T	F	F
T	T	T

• The OR function takes two or more inputs and gives the result of True when any input is True. It is denoted by a “ + “, or sometimes a “ | “.
  Truth Table:

A	B	A + B
F	F	F
F	T	T
T	F	T
T	T	T

• The NOT function takes one input and gives a result opposite of the input. It is denoted by a “ ~ “.
  Truth Table:

A	~A
F	T
T	F