Cardinality of a set

The number of distinct element in a given set A is called the cardinal number of A. It is denoted by n(A). If A = { 1, 2, 3, 4, 5 }, then the cardinality of set A is denoted by n (A) = 5.

Summary

The number of distinct element in a given set A is called the cardinal number of A. It is denoted by n(A). If A = { 1, 2, 3, 4, 5 }, then the cardinality of set A is denoted by n (A) = 5.

Things to Remember

  • The cardinality of a set is a positive integer but it is not decimal. So, n(A)  is not equal to 50% because  50% = 0.5.
  • If A, B and C are disjoint sets, \(n(A \cup B \cup C) = n(A) + n(B) + n(C).\)
  • \((A \cup B \cup C)= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C)\)

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Subjective Questions

Q1:

How waste products are removed from arterial blood in hemodialysis machine. Explain with suitable diagram.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Hemodialysis machine uses the dialyzer for the purification of the blood for the patient with kidney failure problem. Dialyzer or artificial kidney is a meeting point of the two circuits, one in which blood flows &amp; other in which dialysis fluid flows.</p> <p>An artificial kidney is a dialyzing unit which operates outside the patients body. It receives the patients blood from cannulated artery via a plastic tubing. The dialysate is an electrolyte solution of suitable composition and the dialysis takes place across the membrane. The return of the dialyzed blood is by another plastic tube to an appropriate vein.</p> <p>The dialyzing membrane has small perforations which are extremely small and invisible. Waste products in the blood are able to pass through these perforations into the dialysate fluid from where they are immediately washed away. Thus, the waste products are removed from arterial blood in hemodialysis machine.</p> <p></p>

Q2:

What is dialysis?


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Dialysis is the process of removing the waste products from the blood and maintaining the body fluid volume for the proper functioning of the body. Dialysis may help the patient with acute renal kidney failure to recover. In case of patient with chronic renal failure, dialysis is used for a certain period until the suitable donor is available for transplant. It can balance the levels of potassium, chloride, sodium etc in the blood just as natural kidneys. One of the drawback of dialysis is it cannot automatically regulate blood pressure, produce hormones like Erythropoetin etc. One of the type of dialysis that is mostly used is hemodialysis.</p>

Q3:

What is dialyzer? Give the types of the dialyzer available.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Dialyzer is an artificial kidney which provides certainfunction of kidney for the people with kidney failure. It is used for the purification of the blood in the patient having kidney failure. The various types of dialyzer available are:</p> <ol><li>Paralle Plate Dialyzer: They use parallel plate having rigids and grooves in them. They have low resistance which allows the blood to flow sufficiently within them. They have rigid supports which permits negative pressure to be created on dialysate for ultra-filtration of the blood.</li> <li>Kill Dialyzer: It consists of three polypropylene boards with dialyzing membranes laid between them. There is rubber gasket running along the periphery of the boards inner surface to prevent leakage of blood &amp; dialysate. Dialysate enters through the stainless steel port and is distributed to groove running across the end of the board above &amp; below membrane of each layer. The dialysate flows down the longitudinal grooves in the boards before collection &amp; flows out at the opposite end of the board.</li> <li>Coil Dialyzer: It has a tubular membrane placed between flexible support wrapped around a rigid cylindrical core. The coil is immersed in dialyzing bath.</li> <li>Hollow Fibre Dialyzer: It is most commonly used dialyzer among all the available types. It consists of thousand of hollow de-acetylated cellulose diacetate capillaries. The capillaries have internal diameter of 200 microns.</li> </ol>

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Cardinality of a set

Cardinality of a set

Intersection
Intersection

The cardinality of set A is defined as the number of elements in the set A and is denoted by n(A).
For example, if A = {a,b,c,d,e} then cardinality of set A i.e.n(A) = 5

Let A and B are two subsets of a universal set U. Their relation can be shown in Venn-diagram as:

$$ n(A) = n_o( A) + n(A \cap B)$$

$$\text{or,}\: n(A) - n (A \cap B)= n_o(A)$$

$$ n(B) = n_o(B) + n(A \cap B)$$

$$\text {or,}\: n(B) - n(A \cap B) = n_o(B)$$

Also,

\begin{align*} n(A∪B) &= n_o(A) + n(A∩B) + n_o(B)\\ n(A∪B) &= n(A) - n(A∩B) + n(A∩B) + n(B) - n(A∩B)\\ n(A∪B) &= n(A) + n(B)- n(A∩B)\\ \therefore n(A∪B) &= n(A) + n(B) - n(A∩B)\\ \end{align*}

If A and B are disjoint sets then:

\(n(A \cap B) = 0, n(A \cup B) =n(A) + n(B)\)

Again,

\(n(U) = n(A \cup B) + n(\overline {A\cup B)}\)

If \(n(\overline {A \cup B)}\)=0, then \( n(U) = n(A \cup B)\)

Problems involving three sets

Let A, B and C are three non-empty and intersecting sets, then:
\(n(A \cup B \cup C) = n(A) + n(B) +n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C).\)

Intersection

In Venn-diagram

\(n(A)\) = Number of elements in set A.

\(n(B)\) = Number of elements in set B.

\(n(C)\)=Number of element in set C.

\(n_o(A)\) = Number of elements in set A only.

\(n_o(B)\) = Number of elements in set B only.

\(n_o(C)\) = Number of elements in set C only.

\(n_o(A \cap B)\) = Number of elements in set A and B only.

\(n_o(B \cap C)\) = Number of elements in set B and C only.

\(n_o(C \cap A)\) = Number of elements in set A and C only.

\(n(A \cap B \cap C)\) = Number of elements in set A, B and C.

From the Venn-diagram

\begin{align*} n(A \cup B \cup C) &= n_o(A) +n_o(B) +n_o(C) +n_o(A \cap B) +n_o(B \cap C) +n_o(C \cap A) + n(A \cap B \cap C)\\ &= n(A) - n_o(A \cap B) - n_o(C \cap A) - n(A \cap B \cap C) + n(B) - n_o(B \cap C) - n_o(C\cap B) - n(A \cap B \cap C)
+ n(C) - n_o(A \cap C) - n_o(B \cap C) - n(A \cap B \cap C) + n_o(A \cap B) +n_o(B \cap C) +n_o(C \cap A) + n(A \cap B \cap C)\\ &= n(A) + n(B) + n(C) - [n_o(A \cap B) +n(A \cap B \cap C)] - [n_o(A \cap B) +n(A \cap B \cap C)] - [n_o(B \cap C) +n(A \cap B \cap C)] - [n_o(C \cap A) +n(A \cap B \cap C)]+n(A \cap B \cap C)\\ &= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(A \cap C) +n(A \cap B \cap C)\\ \end{align*}

$$\boxed{\therefore (A \cup B \cup C)= n(A) + n(B) + n(C) - n(A \cap B) -n(B \cap C) -n(C \cap A) +n(A \cap B \cap C)} $$

If A, B and C are disjoint sets,

\(n(A \cup B \cup C) = n(A) + n(B) + n(C)\)

Lesson

Sets

Subject

Compulsory Mathematics

Grade

Grade 10

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