Class 11th mathematics (sets)

-
Sets are defined as a well-defined collection of objects. A set without any element is termed as an empty set. A set comprising of definite elements is termed as a finite set whereas if the set has an indefinite number of elements it is termed an infinite set. Two sets P and Q are equal if they have exactly the same number of elements. A set P is a subset of a set Q if all the elements of P are also an element of Q. A power set [A(P)] of a set P comprising of all subsets of P. The union of sets P and Q is a set comprising of all elements which are either in sets P or Q. The intersection of sets P and Q is a set comprising of all common elements of sets P and Q. Similarly, the difference of sets P and Q in the same order is a set comprising of elements belonging to P but not Q.
Important Equations
  1. For any two sets P and Q,
  • (P ∪ Q)′ = P′ ∩ Q′
  • (P ∩ Q)′ = P′ ∪ Q′
  1. If P and Q are finite sets such that P ∩ Q = φ, then n (P ∪ Q) = n (P) + n (Q).
  2. If P ∩ Q ≠ φ, then
n (P ∪ Q) = n (P) + n (Q) – n (P ∩ Q)
  1. n (P ∪ Q ∪ R) = n(P) + n(Q) + n(R) – n(P ∩ Q) – n(P ∩ Q) – n(P ∩ Q ) + n(P ∩ Q ∩ R)
  2. If P is a subset of set U (Universal Set), then its complement (P′) is also a subset of Universal Set (U).

Some Properties of Operation of Intersection

  • P ∩ Q = Q ∩ P (Commutative law).
  • (P ∩ Q) ∩ R = P ∩ (Q ∩ R) (Associative law).
  • φ ∩ P = φ, U ∩ P = P.
  • P ∩ P = P (Idempotent law).
  • P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ Q) (Distributive law).

Some Properties of the Operation of Union

  • P ∪ Q = Q ∪ P (Commutative law).
  • (P ∪ Q) ∪ R = P ∪ ( Q ∪ R) (Associative law).
  • P ∪ φ = P (Law of the identity element).
  • U ∪ P = U (Law of U).

Comments

Post a Comment

Popular posts from this blog

Class 11th science chemistry (thermodynamics)

-
Image
Thermodynamics deals with the change of energy in physical and chemical processes. It allows us to study these changes quantitatively as well as to make effective predictions. Therefore, we divide the universe as surrounding and system. Physical or chemical processes lead to progression or immersion of heat part of it may be converted to work. According to the first law of thermodynamics , these quantities are related as ∆U = q + w where q is heat and w is work. Change in internal energy is a state function and depends on only the initial and final states. On the other hand, w and q are not state functions. The change in temperature due to the transfer of heat one system to another system can be measured. Heat evolved or absorbed is expressed as q = C∆T. In case of expansion of gas, work is measured as w = –p ex ∆V Enthalpy Change Enthalpy change is defined as the change in energy of a system at constant pressure. It is denoted as ΔH. Enthalpy is a state function. Enthalpy ch...
Read more

Class 11th science chemistry (Redox reaction)

-
Image
Redox reaction is the combination of both oxidation and reduction reactions. When redox reaction takes place, both oxidation and reduction reactions also occur simultaneously. What is a Redox Reaction? Redox is a term used for the oxidation-reduction reaction. Redox reaction is a chemical reaction where a change in the oxidation state of atoms occurs. It involves electron transfer i.e if one chemical species gains electrons, then another chemical species gives or loses electrons. The species from which the electron is lost is said to be oxidized whereas the species to which the electron is added is said to be reduced. Example zinc displaces copper in an aqueous solution called copper sulfate. Redox reactions can be classified into four different categories: Combination reaction Decomposition reaction Displacement reaction Disproportionation reaction Oxidizing And Reducing Agents The role of an oxidizing agent (Oxidant) in a chemical reaction is to gain electrons. ...
Read more