Set Theory and Logic is branch of mathematics which concerns with the study of sets. Sets are the collection of objects or data which are known as elements of the set(s).
A is said to be a subset of B if each element of A is an element of B. we denote it as: A ⊂ B.
There are different binary operations applied on two sets as follows:
Union of sets A and B (A∪B) gives all the elements which are either in A or B. eg: if A={4,5,6,7} and B={12,3,4,7} then A∪B is {3,4,5,6,7,12}.
Intersection of sets A and B gives the elements which are common in both sets. Eg:if A={4,5,6,7} and B={12,3,4,7} then A∩B={4,7}.
Difference of sets A and B give the elements of A that are not in B. Eg:if A={4,5,6,7} and B={12,3,4,7} then A-B={5,6}.
Symmetric difference (A∆B)of sets A and B gives the elements which are member of either of the sets but not of both. Eg:if A={4,5,6,7} and B={12,3,4,7} then {3,5,6,12} is symm. diff.
Set Theory Venn Diagram is a diagram which projects all possible relationships among some sets. Sets in a diagram are represented by a closed curves, normally circle in a plane. Above operations can also be represented through these diagrams as shown:
(A-B)
Set Theory Formulas
A∪A=A
A∩A=A
A∪B=B∪A
A∩B= B∩A
(A∪B)∪C=A∪(B∪C)
(A∩B)∩C=A∩(B∩C)
(A∪B)C= AC∩BC
(A∩B)C=AC∪BC
E(A∪B)=E(A)+E(B)–E(A∩B)
E(A∪B∪C)=E(A)+E(B)+E(C)–E(A∩B)–E(A∩C)–E(B∩C)+E(A∩B∩C)
Infinite Set Theory deals with infinite sets. Infinite sets are those which are not finite or the number of elements is infinite. These sets are countable or uncountable. Eg: set of positive integers is countable set while that of real numbers is uncountable.
Probability Set Theory explains probability of outcome of an experiment. The possible outcomes of an experiment are represented using sets. For example when a coin is tossed its possible outcomes are: head and tail. So outcome is given as: S ={head, tail} or {H, T}. Similarly, if two coins are tossed then possible outcomes are given as: {HH, HT, TH, HH}.
The formulas of sets theory are also applicable here. If A and B are two events then P(A∪B)=P(A)+P(B)-P(A∩B). Here P is probability of an event to occur.
Set Theory Proofs Examples to understand this topic clearly.
Example) Let P, Q and R are three sets. If P ∈ Q and Q ⊂ R, then P ⊂ R is true or false? If not true, give example.
Solution)
False. Let P={3}, Q={{3}, 2} and R={{3}, 2, 4}. Here P∈Q as P={3}
and Q⊂R. But P⊄R as 3∈P and 3∉R.
A is said to be a subset of B if each element of A is an element of B. we denote it as: A ⊂ B.
There are different binary operations applied on two sets as follows:
Union of sets A and B (A∪B) gives all the elements which are either in A or B. eg: if A={4,5,6,7} and B={12,3,4,7} then A∪B is {3,4,5,6,7,12}.
Intersection of sets A and B gives the elements which are common in both sets. Eg:if A={4,5,6,7} and B={12,3,4,7} then A∩B={4,7}.
Difference of sets A and B give the elements of A that are not in B. Eg:if A={4,5,6,7} and B={12,3,4,7} then A-B={5,6}.
Symmetric difference (A∆B)of sets A and B gives the elements which are member of either of the sets but not of both. Eg:if A={4,5,6,7} and B={12,3,4,7} then {3,5,6,12} is symm. diff.
Set Theory Venn Diagram is a diagram which projects all possible relationships among some sets. Sets in a diagram are represented by a closed curves, normally circle in a plane. Above operations can also be represented through these diagrams as shown:
(A-B)
Set Theory Formulas
A∪A=A
A∩A=A
A∪B=B∪A
A∩B= B∩A
(A∪B)∪C=A∪(B∪C)
(A∩B)∩C=A∩(B∩C)
(A∪B)C= AC∩BC
(A∩B)C=AC∪BC
E(A∪B)=E(A)+E(B)–E(A∩B)
E(A∪B∪C)=E(A)+E(B)+E(C)–E(A∩B)–E(A∩C)–E(B∩C)+E(A∩B∩C)
Infinite Set Theory deals with infinite sets. Infinite sets are those which are not finite or the number of elements is infinite. These sets are countable or uncountable. Eg: set of positive integers is countable set while that of real numbers is uncountable.
Probability Set Theory explains probability of outcome of an experiment. The possible outcomes of an experiment are represented using sets. For example when a coin is tossed its possible outcomes are: head and tail. So outcome is given as: S ={head, tail} or {H, T}. Similarly, if two coins are tossed then possible outcomes are given as: {HH, HT, TH, HH}.
The formulas of sets theory are also applicable here. If A and B are two events then P(A∪B)=P(A)+P(B)-P(A∩B). Here P is probability of an event to occur.
Set Theory Proofs Examples to understand this topic clearly.
Example) Let P, Q and R are three sets. If P ∈ Q and Q ⊂ R, then P ⊂ R is true or false? If not true, give example.
Solution)
False. Let P={3}, Q={{3}, 2} and R={{3}, 2, 4}. Here P∈Q as P={3}
and Q⊂R. But P⊄R as 3∈P and 3∉R.
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