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Cardinality of Union of Two Sets

Topic: Set Theory BasicsChapter: Chapter 7
n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)
This formula calculates the total number of elements in the union of two sets, A and B. It accounts for elements present in A, elements present in B, and subtracts the elements common to both (intersection) to avoid double-counting. This is used when finding the count of items belonging to at least one of two categories.

Cardinality of Union of Three Sets

Topic: Set Theory BasicsChapter: Chapter 7
n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)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)
This formula extends the concept of cardinality to three sets, A, B, and C. It sums the individual cardinalities, subtracts the cardinalities of pairwise intersections (to correct for double-counting), and then adds back the cardinality of the intersection of all three sets (because elements common to all three were subtracted three times and added three times, effectively removed). This is vital for problems involving three overlapping categories.

Cardinality of Complement of a Set

Topic: Set Theory BasicsChapter: Chapter 7
n(A′)=n(U)−n(A)n(A') = n(U) - n(A)
This formula calculates the number of elements not in set A, given a universal set U. The complement of A (denoted A') includes all elements in the universal set U that are not in A. This is used to find the count of items that do not possess a certain characteristic.

Cardinality of Difference of Two Sets (A only)

Topic: Set Theory BasicsChapter: Chapter 7
n(A−B)=n(A)−n(A∩B)n(A - B) = n(A) - n(A \cap B)
This formula calculates the number of elements that are in set A but not in set B. It represents the elements unique to set A. This is useful when finding the count of items belonging exclusively to one category.

De Morgan's Law (Union Complement)

Topic: Set Theory BasicsChapter: Chapter 7
(A∪B)′=A′∩B′(A \cup B)' = A' \cap B'
This law states that the complement of the union of two sets is equal to the intersection of their complements. In terms of cardinality, this means the number of elements not in either A or B is equal to the number of elements not in A AND not in B. It helps simplify complex set expressions.

De Morgan's Law (Intersection Complement)

Topic: Set Theory BasicsChapter: Chapter 7
(A∩B)′=A′∪B′(A \cap B)' = A' \cup B'
This law states that the complement of the intersection of two sets is equal to the union of their complements. In terms of cardinality, this means the number of elements not common to both A and B is equal to the number of elements not in A OR not in B. It is another tool for simplifying set operations.

Cardinality of Union of Two Disjoint Sets

Topic: Set Theory BasicsChapter: Chapter 7
n(A∪B)=n(A)+n(B)if A∩B=∅n(A \cup B) = n(A) + n(B) \quad \text{if } A \cap B = \emptyset
Calculates the number of elements in the union of two disjoint sets A and B (sets with no common elements). n(A) is the number of elements in set A, and n(B) in set B.

Cardinality of Difference of Two Sets

Topic: Set Theory BasicsChapter: Chapter 7
n(A−B)=n(A)−n(A∩B)n(A - B) = n(A) - n(A \cap B)
Calculates the number of elements in the set difference A - B (elements in A but not in B). n(A) is the number of elements in set A, and n(A \cap B) is the number of elements in the intersection of A and B.

Number of Subsets

Topic: Set Theory BasicsChapter: Chapter 7
2n2^n
Calculates the total number of subsets that can be formed from a set containing n distinct elements.

Number of Proper Subsets

Topic: Set Theory BasicsChapter: Chapter 7
2n−12^n - 1
Calculates the total number of proper subsets that can be formed from a set containing n distinct elements. A proper subset excludes the set itself.

Cardinality of Union of Two Sets

Topic: Logical ReasoningChapter: Chapter 7
n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)
This formula calculates the number of elements in the union of two sets A and B. It sums the elements in A and B, then subtracts the elements common to both (intersection) to avoid double-counting.

Cardinality of Union of Three Sets

Topic: Logical ReasoningChapter: Chapter 7
n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)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)
This formula extends the concept to three sets, calculating the total number of elements in their union by adding individual cardinalities, subtracting pairwise intersections, and adding back the triple intersection.

Number of Subsets

Topic: Logical ReasoningChapter: Chapter 7
2n2^n
If a set has 'n' elements, this formula gives the total number of possible subsets, including the empty set and the set itself.

Number of Proper Subsets

Topic: Logical ReasoningChapter: Chapter 7
2n−12^n - 1
This formula calculates the number of proper subsets, which are all subsets excluding the set itself.

De Morgan's Law for Union

Topic: Logical ReasoningChapter: Chapter 7
(A∪B)′=A′∩B′(A \cup B)' = A' \cap B'
The complement of the union of two sets is equal to the intersection of their complements.

De Morgan's Law for Intersection

Topic: Logical ReasoningChapter: Chapter 7
(A∩B)′=A′∪B′(A \cap B)' = A' \cup B'
The complement of the intersection of two sets is equal to the union of their complements.

Difference of Sets

Topic: Logical ReasoningChapter: Chapter 7
A−B=A∩B′A - B = A \cap B'
The set difference A - B contains elements that are in A but not in B. It can also be expressed as the intersection of A and the complement of B.

Symmetric Difference of Sets

Topic: Logical ReasoningChapter: Chapter 7
AΔB=(A−B)∪(B−A)=(A∪B)−(A∩B)A \Delta B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)
The symmetric difference contains elements that are in A or B but not in their intersection.

Number of Relations from A to B

Topic: Logical ReasoningChapter: Chapter 7
2n(A)×n(B)2^{n(A) \times n(B)}
If set A has n(A) elements and set B has n(B) elements, the number of possible relations from A to B is 2 raised to the power of the number of elements in the Cartesian product A x B.

Number of Reflexive Relations on a Set A

Topic: Logical ReasoningChapter: Chapter 7
2n2−n2^{n^2 - n}
For a set A with 'n' elements, this formula calculates the number of reflexive relations possible on A.

Number of Functions from A to B

Topic: Logical ReasoningChapter: Chapter 7
n(B)n(A)n(B)^{n(A)}
If set A has n(A) elements and set B has n(B) elements, the number of possible functions from A to B is the cardinality of B raised to the power of the cardinality of A.

Number of One-to-One (Injective) Functions from A to B

Topic: Logical ReasoningChapter: Chapter 7
P(n(B),n(A))=n(B)!(n(B)−n(A))! (if n(A)≤n(B), else 0)P(n(B), n(A)) = \frac{n(B)!}{(n(B) - n(A))!}\text{ (if } n(A) \le n(B) \text{, else 0)}
This formula calculates the number of injective functions from A to B, where each element of A maps to a unique element of B. It's applicable only if the number of elements in A is less than or equal to the number of elements in B. If n(A) > n(B), the number of one-to-one functions is 0.

Number of Bijective Functions from A to B

Topic: Logical ReasoningChapter: Chapter 7
n! (if n(A)=n(B)=n, else 0)n!\text{ (if } n(A) = n(B) = n \text{, else 0)}
If n(A) = n(B) = n, the number of bijective (one-to-one and onto) functions from A to B is n factorial. If n(A)
eq n(B), the number of bijective functions is 0.

Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 7
SI=P×R×T100SI = \frac{P \times R \times T}{100}
This formula calculates the interest earned only on the initial principal amount. P is the Principal amount, R is the Annual Interest Rate (in percentage), and T is the Time period (in years). The total amount at the end of the period would be P + SI. This can be viewed as a linear function of time.

Amount (Future Value) with Simple Interest

Topic: Time Value of MoneyChapter: Chapter 7
A=P(1+R×T100)A = P(1 + \frac{R \times T}{100})
This formula calculates the total amount (Principal + Simple Interest) at the end of the period. A is the Amount, P is the Principal, R is the Annual Interest Rate (in percentage), and T is the Time period (in years). This represents the future value when interest is calculated simply.

Compound Interest (Future Value)

Topic: Time Value of MoneyChapter: Chapter 7
FV=P(1+rn)ntFV = P(1 + \frac{r}{n})^{nt}
This formula calculates the future value (FV) of an investment or loan when interest is compounded. P is the Principal, r is the Annual Nominal Interest Rate (as a decimal), n is the number of times interest is compounded per year, and t is the Time period (in years). This formula demonstrates an exponential functional relationship between future value and time/compounding periods.

Present Value (PV) of a Future Sum

Topic: Time Value of MoneyChapter: Chapter 7
PV=FV(1+rn)ntPV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
This formula calculates the present value (PV) of a future sum (FV) that will be received or paid at a later date. FV is the Future Value, r is the Annual Nominal Interest Rate (as a decimal), n is the number of times interest is compounded per year, and t is the Time period (in years). This is the inverse operation of compounding, known as discounting.

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 7
EAR=(1+rn)n−1EAR = (1 + \frac{r}{n})^n - 1
This formula calculates the actual annual rate of return or interest paid, considering the effect of compounding. r is the Annual Nominal Interest Rate (as a decimal), and n is the number of times interest is compounded per year. It allows for comparison of different interest rates with varying compounding frequencies.

Future Value of an Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 7
FVOA=PMT×[(1+i)n−1i]FVOA = PMT \times \left[ \frac{(1 + i)^n - 1}{i} \right]
This formula calculates the future value of a series of equal payments (PMT) made at the end of each period, where i is the interest rate per period (r/n) and n is the total number of payments/periods. An annuity can be seen as a sequence of cash flows, and its future value is a function of the payment amount, interest rate, and number of periods.

Present Value of an Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 7
PVOA=PMT×[1−(1+i)−ni]PVOA = PMT \times \left[ \frac{1 - (1 + i)^{-n}}{i} \right]
This formula calculates the present value of a series of equal payments (PMT) made at the end of each period, where i is the interest rate per period (r/n) and n is the total number of payments/periods. This is crucial for valuing regular income streams or loan repayments.

Present Value of a Perpetuity

Topic: Time Value of MoneyChapter: Chapter 7
PVPerpetuity=PMTiPV_{Perpetuity} = \frac{PMT}{i}
This formula calculates the present value of an infinite series of equal payments (PMT) that start at the end of the first period, where i is the interest rate per period (r/n). It's a special case of an annuity where n approaches infinity.

Simple Interest

Topic: Time Value of MoneyChapter: Chapter 7
SI=P×r×nSI = P \times r \times n
Calculates the simple interest earned on a principal amount. P is the principal, r is the annual interest rate (as a decimal), n is the number of years.

Simple Amount (Future Value with Simple Interest)

Topic: Time Value of MoneyChapter: Chapter 7
FVSI=P(1+r×n)FV_{SI} = P (1 + r \times n)
Calculates the total amount (Principal + Simple Interest) after n years. P is the principal, r is the annual interest rate (as a decimal), n is the number of years.

Compound Amount / Future Value (Annually Compounded)

Topic: Time Value of MoneyChapter: Chapter 7
FV=P(1+r)nFV = P (1 + r)^n
Calculates the future value of a principal amount P compounded annually for n periods at an annual interest rate r (as a decimal).

Compound Amount / Future Value (Compounded m times a year)

Topic: Time Value of MoneyChapter: Chapter 7
FV=P(1+rm)n×mFV = P \left(1 + \frac{r}{m}\right)^{n \times m}
Calculates the future value of a principal amount P compounded m times a year for n years at an annual interest rate r (as a decimal).

Compound Interest

Topic: Time Value of MoneyChapter: Chapter 7
CI=FV−P=P[(1+rm)n×m−1]CI = FV - P = P \left[ \left(1 + \frac{r}{m}\right)^{n \times m} - 1 \right]
Calculates the total compound interest earned on a principal amount P over n years, compounded m times a year at an annual interest rate r (as a decimal).

Present Value (PV) of a Single Sum

Topic: Time Value of MoneyChapter: Chapter 7
PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}
Calculates the present value PV of a future sum FV to be received after n periods, discounted at an annual interest rate r (as a decimal).

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 7
FVAD=A[(1+r)n−1r](1+r)FVAD = A \left[ \frac{(1 + r)^n - 1}{r} \right] (1 + r)
Calculates the future value of a series of n equal payments A made at the beginning of each period, earning an interest rate r (as a decimal) per period.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 7
PVAD=A[1−(1+r)−nr](1+r)PVAD = A \left[ \frac{1 - (1 + r)^{-n}}{r} \right] (1 + r)
Calculates the present value of a series of n equal payments A made at the beginning of each period, discounted at an interest rate r (as a decimal) per period.

Sinking Fund Payment

Topic: Time Value of MoneyChapter: Chapter 7
A=FV[r(1+r)n−1]A = FV \left[ \frac{r}{(1 + r)^n - 1} \right]
Calculates the periodic payment A required to accumulate a future sum FV over n periods at an interest rate r (as a decimal) per period. This is the reciprocal of the FVOA factor.

Loan Amortization Payment / Capital Recovery Factor

Topic: Time Value of MoneyChapter: Chapter 7
A=PV[r1−(1+r)−n]A = PV \left[ \frac{r}{1 - (1 + r)^{-n}} \right]
Calculates the equal periodic payment A required to amortize a loan (present value PV) over n periods at an interest rate r (as a decimal) per period. This is the reciprocal of the PVOA factor.
Chapter 7 Formulas — CA Foundation | Kartavvya