Binary operation | Jamb Mathematics

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Binary operation? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Binary operation together and move one step closer to achieving your exam success! Blissful learning.

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Closure Property Questions

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1. Check if the set of natural numbers $(\mathbb{N})$ is closed under subtraction.

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Solution:

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A set is closed under an operation if performing the operation on any two elements of the set results in an element still in the set.

Let $a, b \in \mathbb{N}$ .
Subtraction: $a - b$

Example:

If $a = 5$ , $b = 3$ , then $5 - 3 = 2 \in \mathbb{N}$ (Valid)
If $a = 3$ , $b = 5$ , then $3 - 5 = -2 \notin \mathbb{N}$ (Invalid)

Since subtraction sometimes results in a number not in $\mathbb{N}$ , $\mathbb{N}$ is not closed under subtraction.

Answer: Not closed under subtraction.

2. Is the set of whole numbers $(\mathbb{W})$ closed under division?

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Solution:

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Whole numbers

(\mathbb{W})

are $0, 1, 2, 3, 4, \dots$ .

Division: $a \div b$
If $a = 4$ , $b = 2$ , then $4 \div 2 = 2 \in \mathbb{W}$ (Valid)
If $a = 5$ , $b = 2$ , then $5 \div 2 = 2.5 \notin \mathbb{W}$ (Invalid)

Since division does not always produce a whole number, $\mathbb{W}$ is not closed under division.

Answer: Not closed under division.

3. Show that the set of integers $(\mathbb{Z})$ is closed under addition.

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Solution:

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For any two integers

a, b \in \mathbb{Z}

Addition: $a + b$

Examples:

If $a = -3$ , $b = 5$ , then $-3 + 5 = 2 \in \mathbb{Z}$
If $a = -7$ , $b = -2$ , then $-7 + (-2) = -9 \in \mathbb{Z}$

Since adding any two integers always results in another integer, $\mathbb{Z}$ is closed under addition.

Answer: Closed under addition.

Commutativity Questions

4. Show that multiplication of real numbers $(\mathbb{R})$ is commutative.

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Solution:

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A binary operation

\star

is commutative if:

a \star b = b \star a

For multiplication:

a \times b = b \times a

Examples:

If $a = 4$ , $b = 7$ , then $4 \times 7 = 28$ and $7 \times 4 = 28$ .
If $a = -3$ , $b = 2$ , then $-3 \times 2 = -6$ and $2 \times (-3) = -6$ .

Since multiplication always satisfies

a \times b = b \times a

, multiplication is commutative in $\mathbb{R}$ .

Answer: Commutative.

5. Is subtraction commutative for integers $(\mathbb{Z})$ ?

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Solution:

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Subtraction is commutative if:

a - b = b - a

Examples:

If $a = 6$ , $b = 3$ , then $6 - 3 = 3$ but $3 - 6 = -3$ (Not Equal)
If $a = -5$ , $b = -2$ , then $-5 - (-2) = -3$ but $-2 - (-5) = 3$ (Not Equal)

Since

a - b \neq b - a

in general, subtraction is not commutative.

Answer: Not commutative.

Associativity Questions

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6. Prove that addition is associative in real numbers $(\mathbb{R})$ .

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Solution:

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A binary operation is associative if:

(a \star b) \star c = a \star (b \star c)

For addition:

(a + b) + c = a + (b + c)

Examples:

Let $a = 2, b = 3, c = 4$ : $(2 + 3) + 4 = 5 + 4 = 9$ $2 + (3 + 4) = 2 + 7 = 9$

Since both expressions are equal, addition is associative.

Answer: Associative.

7. Is division associative in $\mathbb{R}$ ?

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Solution:

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Division is associative if:

(a \div b) \div c = a \div (b \div c)

Let

a = 8, b = 4, c = 2

(8 \div 4) \div 2 = 2 \div 2 = 1

8 \div (4 \div 2) = 8 \div 2 = 4

Since

1 \neq 4

, division is not associative.

Answer: Not associative.

Distributivity Questions

8. Prove that multiplication distributes over addition in $\mathbb{R}$ .

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Solution:

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Multiplication is distributive over addition if:

a \times (b + c) = (a \times b) + (a \times c)

Examples:

Let $a = 2, b = 3, c = 4$ : $2 \times (3 + 4) = 2 \times 7 = 14$ $(2 \times 3) + (2 \times 4) = 6 + 8 = 14$

Since both sides are equal, multiplication distributes over addition.

Answer: Distributive.

9. Is division distributive over addition?

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Solution:

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For distributivity:

a \div (b + c) = (a \div b) + (a \div c)

Let

a = 12, b = 4, c = 2

12 \div (4 + 2) = 12 \div 6 = 2

(12 \div 4) + (12 \div 2) = 3 + 6 = 9

Since

2 \neq 9

, division is not distributive over addition.

Answer: Not distributive.

10. Verify if exponentiation is distributive over multiplication.

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Solution:

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Exponentiation is distributive if:

(a \times b)^c = (a^c) \times (b^c)

Examples:

If $a = 2, b = 3, c = 2$ : $(2 \times 3)^2 = 6^2 = 36$ $(2^2) \times (3^2) = 4 \times 9 = 36$

Since both are equal, exponentiation distributes over multiplication.

Answer: Distributive.

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Closure Property Questions

11. Is the set of rational numbers $(\mathbb{Q})$ closed under division?

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Solution:

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Rational numbers

\mathbb{Q}

are fractions of integers where the denominator is nonzero.

If $a = \frac{3}{4}$ and $b = \frac{2}{5}$ , $a \div b = \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} \in \mathbb{Q}$
If $a = \frac{5}{6}$ and $b = 0$ , then $a \div 0$ is undefined.

Since division by zero is not allowed, $\mathbb{Q}$ is not closed under division.

Answer: Not closed under division.

12. Show that the set of even integers is closed under addition.

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Solution:

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Let

a = 2m

and

b = 2n

, where

m, n

are integers.

a + b = 2m + 2n = 2(m+n)

Since

2(m+n)

is always even, the sum of any two even numbers is also even.

Answer: Closed under addition.

13. Is the set of odd integers closed under multiplication?

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Solution:

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Let

a = 2m + 1

and

b = 2n + 1

, where

m, n

are integers.

a \times b = (2m + 1) \times (2n + 1)

Expanding:

= 4mn + 2m + 2n + 1

= 2(2mn + m + n) + 1

Since this expression is odd, the product of two odd numbers is always odd.

Answer: Closed under multiplication.

Check if the set of prime numbers is closed under subtraction.**
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Solution: Consider prime numbers $p_1 = 5$ and $p_2 = 3$ .

5 - 3 = 2 \in \mathbb{P}

However, for

p_1 = 7

and

p_2 = 5

7 - 5 = 2 \in \mathbb{P}

But for

p_1 = 11

and

p_2 = 7

11 - 7 = 4 \notin \mathbb{P}

Since subtraction can result in a non-prime, the set of prime numbers is not closed under subtraction.

Answer: Not closed under subtraction.

Commutativity Questions

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15. Is exponentiation commutative for real numbers?

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Solution:

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Exponentiation is commutative if:

a^b = b^a

Counterexample:

If $a = 2, b = 3$ , then $2^3 = 8$ and $3^2 = 9$ .

Since

8 \neq 9

, exponentiation is not commutative.

Answer: Not commutative.

16. Show that set intersection $(\cap)$ is commutative.

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Solution: Intersection is commutative if:

A \cap B = B \cap A

Example:

If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$ ,

A \cap B = \{2, 3\} = B \cap A

Since the order does not matter, set intersection is commutative.

Answer: Commutative.

17. Is matrix multiplication commutative?

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Solution:

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Matrix multiplication is commutative if:

A \times B = B \times A

Counterexample: Let

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

and

B = \begin{bmatrix} 3 & 0 \\ 4 & 5 \end{bmatrix}

A \times B \neq B \times A

Since matrix multiplication does not always commute, it is not commutative.

Answer: Not commutative.

Associativity Questions

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#18. Show that multiplication is associative in real numbers $(\mathbb{R})$ .

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Solution:

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Multiplication is associative if:

(a \times b) \times c = a \times (b \times c)

Example:

If $a = 2, b = 3, c = 4$ :

(2 \times 3) \times 4 = 6 \times 4 = 24

2 \times (3 \times 4) = 2 \times 12 = 24

Since both are equal, multiplication is associative.

Answer: Associative.

19. Is exponentiation associative?

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Solution:

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Exponentiation is associative if:

(a^b)^c = a^{(b^c)}

Counterexample:

Let $a = 2, b = 3, c = 2$ :

(2^3)^2 = 8^2 = 64

2^{(3^2)} = 2^9 = 512

Since

64 \neq 512

, exponentiation is not associative.

Answer: Not associative.

Distributivity Questions

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20. Verify if modulus operation $(\mod)$ distributes over addition.

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Solution: Distributivity means:

(a + b) \mod m = (a \mod m + b \mod m) \mod m

Example:

Let $a = 7, b = 5, m = 4$ :

(7 + 5) \mod 4 = 12 \mod 4 = 0

(7 \mod 4 + 5 \mod 4) \mod 4 = (3 + 1) \mod 4 = 4 \mod 4 = 0

Since both are equal, modulus distributes over addition.

Answer: Distributive.

21. Is set union $(\cup)$ distributive over intersection $(\cap)$ ?

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Solution:

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Distributivity means:

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

Example:

If $A = \{1, 2\}$ , $B = \{2, 3\}$ , $C = \{2, 4\}$ :

B \cap C = \{2\}

A \cup (B \cap C) = \{1, 2\}

(A \cup B) = \{1, 2, 3\}, (A \cup C) = \{1, 2, 4\}

(A \cup B) \cap (A \cup C) = \{1, 2\}

Since both sides are equal, set union distributes over intersection.

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Calculation problems involving identity and inverse element

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Identity Element Questions

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1. Find the identity element for addition in $\mathbb{R}$ (set of real numbers).

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Solution:

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The identity element

e

for an operation

\star

satisfies:

a \star e = e \star a = a

For addition:

a + e = a

Solving for

e

e = 0

Answer: The identity element for addition in $\mathbb{R}$ is 0.

2. Find the identity element for multiplication in $\mathbb{R}$ .

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Solution:

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For multiplication:

a \times e = e \times a = a

Solving for

e

e = 1

Answer: The identity element for multiplication in $\mathbb{R}$ is 1.

3. Is there an identity element for subtraction in $\mathbb{R}$ ?

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Solution:

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For subtraction:

a - e = a

Solving for

e

e = 0

Checking:

a - 0 = a, \quad 0 - a \neq a

Since

e - a \neq a

, subtraction does not have an identity element.

Answer: No identity element for subtraction.

4. Is 0 an identity element for division in $\mathbb{R}$ ?

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Solution:

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For division:

a \div e = a

Setting

e = 0

a \div 0

Since division by zero is undefined, 0 is not an identity element for division.

Answer: No identity element for division.

5. Find the identity element in matrix addition.

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Solution:

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For an

n \times n

matrix

A

, the identity element

I

satisfies:

A + I = A

The identity matrix for addition is the zero matrix

O

O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Answer: The zero matrix is the identity element for matrix addition.

Inverse Element Questions

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6. Find the additive inverse of 7 in $\mathbb{R}$ .

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Solution:

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The additive inverse

a^{-1}

satisfies:

a + a^{-1} = 0

For

a = 7

7 + (-7) = 0

Answer: The additive inverse of 7 is -7.

7. Find the multiplicative inverse of 5 in $\mathbb{R}$ .

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Solution:

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The multiplicative inverse

a^{-1}

satisfies:

a \times a^{-1} = 1

For

a = 5

5 \times \frac{1}{5} = 1

Answer: The multiplicative inverse of 5 is $\frac{1}{5}$ .

8. Find the additive inverse of -12 in $\mathbb{R}$ .

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Solution:**

-12 + 12 = 0

Answer: The additive inverse of -12 is 12.

9. Find the multiplicative inverse of $\frac{3}{7}$ in $\mathbb{Q}$ (set of rational numbers).

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Solution:** The multiplicative inverse is:

\left(\frac{3}{7}\right)^{-1} = \frac{7}{3}

\frac{3}{7} \times \frac{7}{3} = 1

Answer: The multiplicative inverse of $\frac{3}{7}$ is $\frac{7}{3}$ .

10. Does 0 have a multiplicative inverse in $\mathbb{R}$ ?

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Solution:

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A number

a^{-1}

satisfies:

0 \times a^{-1} = 1

Since

0 \times x = 0

for any

x

, no number exists that satisfies $0 \times x = 1$ .

Answer: 0 has no multiplicative inverse.

11. Find the inverse of matrix $A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$ .

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Solution:

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The inverse of a matrix

A

is:

A^{-1} = \frac{1}{\det(A)} \text{adj}(A)

Determinant:

\det(A) = 2 \times 2 - 0 \times 0 = 4

A^{-1} = \frac{1}{4} \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}

= \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{bmatrix}

Answer: $\begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$

12. Find the inverse of 3 in modulo 7 arithmetic.

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Solution:

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The inverse of

a

\mod m

satisfies:

a \times x \equiv 1 \pmod{m}

Finding

x

such that:

3x \equiv 1 \pmod{7}

Checking values:

3 \times 5 = 15 \equiv 1 \pmod{7}

Answer: The inverse of 3 in $\mod 7$ is 5.

13. Find the inverse of -4 in mod 9 arithmetic.

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Solution:

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Finding

x

such that:

-4 \times x \equiv 1 \pmod{9}

Equivalent to solving:

5x \equiv 1 \pmod{9}

Checking values:

5 \times 2 = 10 \equiv 1 \pmod{9}

Answer: The inverse of -4 in mod 9 is 2.

14. Find the inverse of 2 in modulo 5 arithmetic.

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Solution:

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Solving:

2x \equiv 1 \pmod{5}

Checking values:

2 \times 3 = 6 \equiv 1 \pmod{5}

Answer: The inverse of 2 in mod 5 is 3.

15. Find the additive inverse of 9 in mod 11 arithmetic.

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Solution:

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9 + x \equiv 0 \pmod{11}

Solving:

x = -9 \equiv 2 \pmod{11}

Answer: The additive inverse of 9 in mod 11 is 2.

16. Find the inverse of the identity matrix.

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Solution:

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The inverse of the identity matrix

I

satisfies:

I \times I^{-1} = I

Since

I \times I = I

, we get:

I^{-1} = I

Answer: The inverse of the identity matrix is itself.

17. Find the inverse of 4 in mod 7 arithmetic.

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Solution:

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4x \equiv 1 \pmod{7}

Checking values:

4 \times 2 = 8 \equiv 1 \pmod{7}

Answer: The inverse of 4 in mod 7 is 2.

18. Find the additive inverse of 17 in mod 19 arithmetic.

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Solution:

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17 + x \equiv 0 \pmod{19}

Solving:

x = -17 \equiv 2 \pmod{19}

Answer: The additive inverse of 17 in mod 19 is 2.

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This is all we can take on "Jamb Mathematics - Lesson Notes on Binary operation for UTME Candidate"

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Binary operation | Jamb Mathematics

Closure Property Questions

Commutativity Questions

Associativity Questions

Distributivity Questions

Closure Property Questions

Commutativity Questions

Associativity Questions

Distributivity Questions

Calculation problems involving identity and inverse element

Identity Element Questions

Inverse Element Questions

I recommend you check my Post on the following:

POSCHOLARS TEAM.

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