This document discusses several key concepts regarding integers: 1) It defines the division algorithm for integers which uniquely expresses any integer m as a quotient q and remainder r when divided by a nonzero integer n. 2) It defines prime numbers as integers greater than 1 that are only divisible by 1 and themselves. It provides an algorithm for determining if a number is prime. 3) It states that every positive integer can be uniquely expressed as a product of prime numbers raised to powers, known as prime factorization. 4) It defines the greatest common divisor (GCD) and least common multiple (LCM) of two integers and provides the Euclidean algorithm for efficiently calculating GCDs.

1. FUNDAMENTALS

2. Properties of Integers
Theorem: If 𝑛𝑛 and 𝑚𝑚 are integers and 𝑛𝑛 > 0, we can
uniquely write 𝑚𝑚 = 𝑞𝑞 ⋅ 𝑛𝑛 + 𝑟𝑟 for integers 𝑞𝑞 and 𝑟𝑟 with
0 ≤ 𝑟𝑟 < 𝑛𝑛.
𝑟𝑟 = 𝑚𝑚 mod 𝑛𝑛
Example:
 𝑛𝑛 = 3, 𝑚𝑚 = 16: 16 = 5 ⋅ 3 + 1
 𝑛𝑛 = 10, 𝑚𝑚 = 3: 3 = 0 ⋅ 10 + 3
 𝑛𝑛 = 5, 𝑚𝑚 = −11: −11 = (−3) ⋅ 5 + 4
 𝑛𝑛 = 5, 𝑚𝑚 = 10: 10 = 2 ⋅ 5 + 0
2© S. Turaev, CSC 1700 Discrete Mathematics

3. Properties of Integers
 If 𝑟𝑟 = 0, then 𝑚𝑚 = 𝑞𝑞 ⋅ 𝑛𝑛, i.e., 𝑚𝑚 is a multiple of 𝑛𝑛. We
write 𝑛𝑛|𝑚𝑚.
 If 𝑟𝑟 ≠ 0, then 𝑚𝑚 is not a multiple of 𝑛𝑛. We write 𝑛𝑛 ∤ 𝑚𝑚.
Theorem:
 If 𝑎𝑎|𝑏𝑏 and 𝑎𝑎|𝑐𝑐, then 𝑎𝑎|(𝑏𝑏 + 𝑐𝑐).
 If 𝑎𝑎|𝑏𝑏 and 𝑎𝑎|𝑐𝑐, then 𝑎𝑎|(𝑏𝑏 − 𝑐𝑐).
 If 𝑎𝑎|𝑏𝑏 and 𝑎𝑎|𝑐𝑐, then 𝑎𝑎|(𝑏𝑏𝑏𝑏).
 If 𝑎𝑎|𝑏𝑏 and 𝑏𝑏|𝑐𝑐, then 𝑎𝑎|𝑐𝑐.
3© S. Turaev, CSC 1700 Discrete Mathematics

4. Prime Numbers
Definition: A positive integer 𝑝𝑝 > 1 is called prime, if the
only positive integers that divide 𝑝𝑝 are 𝑝𝑝 and 1.
Example: 2, 3, 5, 7, 11, 13, … are prime.
4© S. Turaev, CSC 1700 Discrete Mathematics
Is 1 prime?

5. Prime Numbers
Algorithm for determining if an integer 𝑛𝑛 > 1 is prime:
Step 1: Check if 𝑛𝑛 is 2. if so, 𝑛𝑛 is prime. If not, proceed to
Step 2: Check if 2|𝑛𝑛. if so, 𝑛𝑛 isn’t prime. If not, proceed to
Step 3: Compute the largest integer 𝑘𝑘 ≤ 𝑛𝑛; proceed to
Step 4: Check if 𝑑𝑑|𝑛𝑛 where 𝑑𝑑 is any odd number in (1, 𝑘𝑘].
If 𝑑𝑑|𝑛𝑛, then 𝑛𝑛 isn’t prime; otherwise it is prime.
5© S. Turaev, CSC 1700 Discrete Mathematics

6. Prime Factorization
Theorem: Every positive integer 𝑛𝑛 > 1 can be written
uniquely as
𝑛𝑛 = 𝑝𝑝1
𝑘𝑘1
𝑝𝑝2
𝑘𝑘2
⋯ 𝑝𝑝𝑠𝑠
𝑘𝑘𝑠𝑠
where
 𝑝𝑝1 < 𝑝𝑝2 < ⋯ < 𝑝𝑝𝑠𝑠 are distinct prime numbers that
divide 𝑛𝑛,
 the 𝑘𝑘’s are positive integers giving the number of
times each prime occurs as a factor of 𝑛𝑛.
Example: 9 = 3 ⋅ 3 = 32, 36 =, 100 =
6© S. Turaev, CSC 1700 Discrete Mathematics

7. Greatest Common Divisor
Definition:
 If 𝑎𝑎, 𝑏𝑏 and 𝑘𝑘 are in ℤ+, and 𝑘𝑘|𝑎𝑎 and 𝑘𝑘|𝑏𝑏, we say that
𝑘𝑘 is a common divisor of 𝑎𝑎 and 𝑏𝑏.
 If 𝑑𝑑 is the largest such 𝑘𝑘, 𝑑𝑑 is called the greatest
common divisor of 𝑎𝑎 and 𝑏𝑏, and we write 𝑑𝑑 =
gcd(𝑎𝑎, 𝑏𝑏).
Example:
 the common divisors of 12 and 30: 1, 2, 3, 6.
gcd 12,30 = 6
 gcd 17,95 = 1 (relatively prime numbers)
7© S. Turaev, CSC 1700 Discrete Mathematics

8. Euclidean Algorithm
 Suppose 𝑎𝑎 > 𝑏𝑏 > 0. We can write
𝑎𝑎 = 𝑘𝑘𝑘𝑘 + 𝑟𝑟
where 𝑘𝑘 ∈ ℤ+
and 0 ≤ 𝑟𝑟 < 𝑏𝑏.
 If a number 𝑛𝑛 divides 𝑎𝑎 and 𝑏𝑏, then it must divide 𝑟𝑟,
since 𝑟𝑟 = 𝑎𝑎 − 𝑘𝑘𝑘𝑘. Thus,
gcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)
8© S. Turaev, CSC 1700 Discrete Mathematics

9. Euclidean Algorithm
gcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)
divide 𝑎𝑎 by 𝑏𝑏: 𝑎𝑎 = 𝑘𝑘1 𝑏𝑏 + 𝑟𝑟1 0 ≤ 𝑟𝑟1 < 𝑏𝑏
divide 𝑏𝑏 by 𝑟𝑟1: 𝑏𝑏 = 𝑘𝑘2 𝑟𝑟1 + 𝑟𝑟2 0 ≤ 𝑟𝑟2 < 𝑟𝑟1
divide 𝑟𝑟1 by 𝑟𝑟2: 𝑟𝑟1 = 𝑘𝑘3 𝑟𝑟2 + 𝑟𝑟3 0 ≤ 𝑟𝑟3 < 𝑟𝑟2
…
divide 𝑟𝑟𝑛𝑛−2 by 𝑟𝑟𝑛𝑛−1: 𝑟𝑟𝑛𝑛−2 = 𝑘𝑘𝑛𝑛 𝑟𝑟𝑛𝑛−1 + 𝑟𝑟𝑛𝑛 0 ≤ 𝑟𝑟𝑛𝑛 < 𝑟𝑟𝑛𝑛−1
divide 𝑟𝑟𝑛𝑛−1 by 𝑟𝑟𝑛𝑛: 𝑟𝑟𝑛𝑛−1 = 𝑘𝑘𝑛𝑛+1 𝑟𝑟𝑛𝑛 + 𝑟𝑟𝑛𝑛+1 0 ≤ 𝑟𝑟𝑛𝑛+1 < 𝑟𝑟𝑛𝑛
𝑎𝑎 > 𝑏𝑏 > 𝑟𝑟1 > 𝑟𝑟2 > 𝑟𝑟3 > ⋯
9© S. Turaev, CSC 1700 Discrete Mathematics

10. Euclidean Algorithm
gcd 𝑎𝑎, 𝑏𝑏 = gcd(𝑏𝑏, 𝑎𝑎 mod 𝑏𝑏)
𝑎𝑎 > 𝑏𝑏 > 𝑟𝑟1 > 𝑟𝑟2 > 𝑟𝑟3 > ⋯ > 𝑟𝑟𝑛𝑛+1 = 0
𝑟𝑟𝑛𝑛−1 = 𝑘𝑘𝑛𝑛+1 𝑟𝑟𝑛𝑛 : 𝑟𝑟𝑛𝑛 𝑟𝑟𝑛𝑛−1, 𝑟𝑟𝑛𝑛 𝑟𝑟𝑛𝑛−2, … , 𝑟𝑟𝑛𝑛 𝑟𝑟2, 𝑟𝑟𝑛𝑛 𝑟𝑟1, 𝑟𝑟𝑛𝑛|𝑏𝑏, 𝑟𝑟𝑛𝑛|𝑎𝑎
gcd 𝑎𝑎, 𝑏𝑏 =
gcd 𝑏𝑏, 𝑟𝑟1 = gcd 𝑟𝑟1, 𝑟𝑟2 = ⋯ = gcd 𝑟𝑟𝑛𝑛−1, 𝑟𝑟𝑛𝑛 = 𝑟𝑟𝑛𝑛
Example: Compute gcd(123, 36)
10© S. Turaev, CSC 1700 Discrete Mathematics

11. Least Common Multiple
Definition:
 If 𝑎𝑎, 𝑏𝑏 and 𝑘𝑘 are in ℤ+, and 𝑎𝑎|𝑘𝑘 and 𝑏𝑏|𝑘𝑘, we say that
𝑘𝑘 is a common multiple of 𝑎𝑎 and 𝑏𝑏.
 If 𝑐𝑐 is the smallest such 𝑘𝑘, 𝑐𝑐 is called the least
common multiple of 𝑎𝑎 and 𝑏𝑏, and we write 𝑐𝑐 =
lcm(𝑎𝑎, 𝑏𝑏).
Example:
 the common multiples of 2 and 3: 6, 12, 18, 24, …
lcm 2,3 = 6.
11© S. Turaev, CSC 1700 Discrete Mathematics

12. Least Common Multiple
Theorem: if 𝑎𝑎, 𝑏𝑏 ∈ ℤ+, then
gcd(𝑎𝑎, 𝑏𝑏) ⋅ lcm 𝑎𝑎, 𝑏𝑏 = 𝑎𝑎 ⋅ 𝑏𝑏.
Proof: Use prime factorizations of 𝑎𝑎 and 𝑏𝑏.
Example: Let 𝑎𝑎 = 540 and 𝑏𝑏 = 504. Find gcd and lcm.
12© S. Turaev, CSC 1700 Discrete Mathematics

13. Exercises
• Write 𝑚𝑚 as 𝑞𝑞𝑞𝑞 + 𝑟𝑟, with 0 ≤ 𝑟𝑟 < 𝑛𝑛:
1. 𝑚𝑚 = 20, 𝑛𝑛 = 3
2. 𝑚𝑚 = 64, 𝑛𝑛 = 37
3. 𝑚𝑚 = 3, 𝑛𝑛 = 12
• Write each integer as a product of powers of primes:
828, 1666, 1781
• Find the gcd(𝑎𝑎, 𝑏𝑏) and lcm(𝑎𝑎, 𝑏𝑏):
1. 𝑎𝑎 = 60, 𝑏𝑏 = 100
2. 𝑎𝑎 = 77, 𝑏𝑏 = 128
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14. Representation of Integers (Reading)
• 3245 means the sum of 3 times 103, 2 times 102, 4
times 10 and 5:
3245 = 3 ⋅ 103 + 2 ⋅ 102 + 4 ⋅ 10 + 5 ⋅ 100
• The base 10 expansion or decimal expansion of 3245
• 10 is called the base of the expansion.
Example:
56893 =
14© S. Turaev, CSC 1700 Discrete Mathematics

15. Representation of Integers (Reading)
Theorem: if 𝑏𝑏 ∈ ℤ+, then every positive integer 𝑛𝑛 can be
uniquely expressed in the form
𝑛𝑛 = 𝑑𝑑𝑘𝑘 ⋅ 𝑏𝑏𝑘𝑘 + 𝑑𝑑𝑘𝑘−1 ⋅ 𝑏𝑏𝑘𝑘−1 + ⋯ 𝑑𝑑1 ⋅ 𝑏𝑏 + 𝑑𝑑0
where 0 ≤ 𝑑𝑑𝑖𝑖 < 𝑏𝑏, 𝑖𝑖 = 1,2, … , 𝑘𝑘, and 𝑑𝑑𝑘𝑘 ≠ 0.
The sequence 𝑑𝑑𝑘𝑘 𝑑𝑑𝑘𝑘−1 ⋯ 𝑑𝑑1 𝑑𝑑0 (more explicitly,
𝑑𝑑𝑘𝑘 𝑑𝑑𝑘𝑘−1 ⋯ 𝑑𝑑1 𝑑𝑑0 𝑏𝑏) is called the base 𝑏𝑏 expansion of 𝑛𝑛.
Example: Find the base 2, 3, 4 representations of 173.
Example: Find the decimal expansion of 100111 2.
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16. Matrices
Definition: A matrix is a rectangular array of numbers in
𝑚𝑚 horizontal rows and 𝑛𝑛 vertical columns:
𝐀𝐀 =
𝑎𝑎11 𝑎𝑎12
𝑎𝑎21 𝑎𝑎22
⋯
𝑎𝑎1𝑛𝑛
𝑎𝑎2𝑛𝑛
⋮ ⋮
𝑎𝑎 𝑚𝑚1 𝑎𝑎 𝑚𝑚𝑚 ⋯ 𝑎𝑎 𝑚𝑚𝑚𝑚
Definition: The 𝑖𝑖th row of 𝐀𝐀 is [𝑎𝑎𝑖𝑖 𝑖 𝑎𝑎𝑖𝑖 𝑖 ⋯ 𝑎𝑎𝑖𝑖 𝑖𝑖], 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚,
and the 𝑗𝑗th column of 𝐀𝐀 is
𝑎𝑎1𝑗𝑗
𝑎𝑎2𝑗𝑗
⋮
𝑎𝑎𝑛𝑛𝑛𝑛
, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.
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17. Matrices
Definition: We say that 𝐀𝐀 is 𝑚𝑚 by 𝑛𝑛, written 𝑚𝑚 × 𝑛𝑛. If
𝑚𝑚 = 𝑛𝑛, we say 𝐀𝐀 is a square matrix of order 𝑛𝑛. The
elements 𝑎𝑎11, 𝑎𝑎22, … , 𝑎𝑎𝑛𝑛𝑛𝑛 form main diagonal of 𝐀𝐀.
Definition: We refer to the element 𝑎𝑎𝑖𝑖𝑖𝑖 in the 𝑖𝑖th row and
𝑗𝑗th column of 𝐀𝐀 as the (𝑖𝑖, 𝑗𝑗) entry of 𝐀𝐀. We denote the
matrix 𝐀𝐀 by [𝑎𝑎𝑖𝑖𝑖𝑖], for short.
Example:
𝐀𝐀 =
2 3 5
0 −1 2
, 𝐁𝐁 =
2 6
3 6
, 𝐂𝐂 = 1 0 0 , 𝐃𝐃 =
1
1
1
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18. Matrices
Definition: A square matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] is called a diagonal
matrix if 𝑎𝑎𝑖𝑖𝑖𝑖 = 0 for all 𝑖𝑖 ≠ 𝑗𝑗.
Example:
𝐅𝐅 =
2 0
0 6
, 𝐆𝐆 =
1 0 0
0 3 0
0 0 −3
, 𝐇𝐇 =
0 0 0
0 3 0
0 0 0
Definition: A 𝑚𝑚 × 𝑛𝑛 matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] is called zero matrix,
denoted by 𝟎𝟎, if 𝑎𝑎𝑖𝑖𝑖𝑖 = 0 for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.
18© S. Turaev, CSC 1700 Discrete Mathematics

19. Matrices
Definition: Two 𝑚𝑚 × 𝑛𝑛 matrices 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑖𝑖]
are said to be equal if 𝑎𝑎𝑖𝑖𝑖𝑖 = 𝑏𝑏𝑖𝑖𝑖𝑖 for all 𝑖𝑖 and 𝑗𝑗.
Definition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑖𝑖] are 𝑚𝑚 × 𝑛𝑛 matrices,
then the sum of 𝐀𝐀 and 𝐁𝐁, denoted by 𝐀𝐀 + 𝐁𝐁, is the matrix
𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑖𝑖] defined by 𝑐𝑐𝑖𝑖𝑖𝑖 = 𝑎𝑎𝑖𝑖𝑖𝑖 + 𝑏𝑏𝑖𝑖𝑖𝑖 for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚,
1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.
Example: Find 𝐀𝐀+ 𝐁𝐁 if
𝐀𝐀 =
1 0 −8
3 3 0
−2 9 −3
, 𝐁𝐁 =
4 5 3
0 −3 2
5 0 −2
19© S. Turaev, CSC 1700 Discrete Mathematics

20. Matrices
Theorem:
 𝐀𝐀 + 𝐁𝐁 = 𝐁𝐁 + 𝐀𝐀
 𝐀𝐀 + 𝐁𝐁 + 𝐂𝐂 = 𝐀𝐀 + (𝐁𝐁 + 𝐂𝐂)
 𝐀𝐀 + 𝟎𝟎 = 𝟎𝟎 + 𝐀𝐀 = 𝐀𝐀
20© S. Turaev, CSC 1700 Discrete Mathematics

21. Matrices
Definition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] is 𝑚𝑚 × 𝑝𝑝 matrix and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑖𝑖] are
𝑝𝑝 × 𝑛𝑛 matrix, then the product of 𝐀𝐀 and 𝐁𝐁, denotes by
𝐀𝐀𝐀𝐀 is the 𝑚𝑚 × 𝑛𝑛 matrix 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑖𝑖] defined by
𝑐𝑐𝑖𝑖𝑖𝑖 = 𝑎𝑎𝑖𝑖 𝑖 𝑏𝑏1𝑗𝑗 + 𝑎𝑎𝑖𝑖2 𝑏𝑏2𝑗𝑗 + ⋯ 𝑎𝑎𝑖𝑖 𝑝𝑝 𝑏𝑏𝑝𝑝𝑗𝑗
for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚, 1 ≤ 𝑗𝑗 ≤ 𝑛𝑛.
Example: Find 𝐀𝐀𝐁𝐁 if
𝐀𝐀 =
1 0 −8
3 3 0
−2 9 −3
, 𝐁𝐁 =
4 5 3
0 −3 2
5 0 −2
21© S. Turaev, CSC 1700 Discrete Mathematics

22. Matrices
Theorem:
 𝐀𝐀(𝐁𝐁𝐁𝐁) = 𝐀𝐀𝐀𝐀 𝐂𝐂
 𝐀𝐀 𝐁𝐁 + 𝐂𝐂 = 𝐀𝐀𝐀𝐀 + 𝐀𝐀𝐀𝐀
 𝐀𝐀 + 𝐁𝐁 𝐂𝐂 = 𝐀𝐀𝐀𝐀 + 𝐁𝐁𝐁𝐁
22© S. Turaev, CSC 1700 Discrete Mathematics

23. Matrices
Definition: A 𝑛𝑛 × 𝑛𝑛 diagonal matrix:
𝐈𝐈𝑛𝑛 =
1 0
0 1
⋯
0
0
⋮ ⋮
0 0 ⋯ 1
is called the identity matrix of order 𝑛𝑛.
Theorem: For any 𝑛𝑛 × 𝑛𝑛 matrix 𝐀𝐀 and positive integer 𝑝𝑝,
 𝐈𝐈𝑛𝑛 𝐀𝐀 = 𝐀𝐀𝐈𝐈𝑛𝑛 = 𝐀𝐀
 𝐀𝐀𝑝𝑝
= 𝐀𝐀 ⋅ 𝐀𝐀 ⋅ ⋯ ⋅ 𝐀𝐀
𝑝𝑝
, 𝐀𝐀0
= 𝐈𝐈𝑛𝑛.
23© S. Turaev, CSC 1700 Discrete Mathematics

24. Matrices
Definition: If 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] is 𝑚𝑚 × 𝑛𝑛 matrix, then the 𝑛𝑛 × 𝑚𝑚
matrix 𝐀𝐀𝑇𝑇 = [𝑎𝑎𝑖𝑖𝑖𝑖
𝑇𝑇
], where 𝑎𝑎𝑖𝑖𝑖𝑖
𝑇𝑇
= 𝑎𝑎𝑗𝑗𝑗𝑗, for all 1 ≤ 𝑖𝑖 ≤ 𝑚𝑚,
1 ≤ 𝑗𝑗 ≤ 𝑛𝑛, is called the transpose of 𝐀𝐀.
Example: Find 𝐀𝐀𝑇𝑇 and 𝐁𝐁𝑇𝑇 if
𝐀𝐀 =
1 0 −8
3 3 0
−2 9 −3
, 𝐁𝐁 =
2 −3 5
6 1 3
24© S. Turaev, CSC 1700 Discrete Mathematics

25. Matrices
Theorem:
 𝐀𝐀𝑇𝑇 𝑇𝑇
= 𝐀𝐀
 𝐀𝐀 + 𝐁𝐁 𝑇𝑇 = 𝐀𝐀𝑇𝑇 + 𝐁𝐁𝑇𝑇
 𝐀𝐀𝐁𝐁 𝑇𝑇 = 𝐁𝐁𝑇𝑇 𝐀𝐀𝑇𝑇
Definition: A matrix 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] is called symmetric if
𝑨𝑨𝑇𝑇=𝐀𝐀.
Definition: if 𝐀𝐀 and 𝐁𝐁 are 𝑛𝑛 × 𝑛𝑛 matrices, we say 𝐁𝐁 is the
inverse of 𝐀𝐀 if 𝐀𝐀𝐁𝐁 = 𝐈𝐈𝑛𝑛.
25© S. Turaev, CSC 1700 Discrete Mathematics

26. Boolean Matrix Operations
Definition: A Boolean matrix is 𝑚𝑚 × 𝑛𝑛 matrix whose
entries are either zero or one.
Definition: Let 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] and 𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑖𝑖] be 𝑚𝑚 × 𝑛𝑛 Boolean
matrices. We define 𝐀𝐀 ∨ 𝐁𝐁 = 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑖𝑖], the join of 𝐀𝐀 and
𝐁𝐁, by
𝑐𝑐𝑖𝑖𝑖𝑖 = 1 if 𝑎𝑎𝑖𝑖𝑖𝑖 = 1 or 𝑏𝑏𝑖𝑖𝑖𝑖 = 1, 𝑐𝑐𝑖𝑖𝑖𝑖 = 0 if 𝑎𝑎𝑖𝑖𝑖𝑖 = 𝑏𝑏𝑖𝑖𝑖𝑖 = 0,
and the meet of 𝐀𝐀 and 𝐁𝐁, by
𝑐𝑐𝑖𝑖𝑖𝑖 = 1 if 𝑎𝑎𝑖𝑖𝑖𝑖 = 𝑏𝑏𝑖𝑖𝑖𝑖 = 1, 𝑐𝑐𝑖𝑖𝑖𝑖 = 0 if 𝑎𝑎𝑖𝑖𝑖𝑖 = 0 or 𝑏𝑏𝑖𝑖𝑖𝑖 = 0.
26© S. Turaev, CSC 1700 Discrete Mathematics

27. Boolean Matrix Operations
Example: Compute 𝐀𝐀 ∨ 𝐁𝐁 and 𝐀𝐀 ∧ 𝐁𝐁 if
𝐀𝐀 =
1 0 1
0 1 1
1 1 0
0 0 0
, 𝐁𝐁 =
1 1 0
1 0 1
0 0 1
1 1 0
27© S. Turaev, CSC 1700 Discrete Mathematics

28. Boolean Matrix Operations
Definition: Let 𝐀𝐀 = [𝑎𝑎𝑖𝑖𝑖𝑖] be an 𝑚𝑚 × 𝑝𝑝 Boolean matrix and
𝐁𝐁 = [𝑏𝑏𝑖𝑖𝑖𝑖] be a 𝑝𝑝 × 𝑛𝑛 Boolean matrix. The Boolean
product of 𝐀𝐀 and 𝐁𝐁, denoted by 𝐀𝐀 ⊙ 𝐁𝐁, is the 𝑚𝑚 × 𝑛𝑛
matrix 𝐂𝐂 = [𝑐𝑐𝑖𝑖𝑖𝑖] defined by
 𝑐𝑐𝑖𝑖𝑖𝑖 = 1 if 𝑎𝑎𝑖𝑖𝑖𝑖 = 1 and 𝑏𝑏𝑖𝑖𝑖𝑖 = 1, for some 1 ≤ 𝑘𝑘 ≤ 𝑝𝑝,
 𝑐𝑐𝑖𝑖𝑖𝑖 = 0 otherwise.
28© S. Turaev, CSC 1700 Discrete Mathematics

29. Boolean Matrix Operations
Example: Compute 𝐀𝐀 ⊙ 𝐁𝐁 if
𝐀𝐀 =
1 0 1
0 1 1
1 1 0
, 𝐁𝐁 =
1 1 0
1 0 1
0 0 1
29© S. Turaev, CSC 1700 Discrete Mathematics

30. Boolean Matrix Operations
Theorem:
 𝐀𝐀 ∨ 𝐁𝐁 = 𝐁𝐁 ∨ 𝐀𝐀
 𝐀𝐀 ∧ 𝐁𝐁 = 𝐁𝐁 ∧ 𝐀𝐀
 𝐀𝐀 ∨ 𝐁𝐁 ∨ 𝐂𝐂 = 𝐀𝐀 ∨ (𝐁𝐁 ∨ 𝐂𝐂)
 𝐀𝐀 ∧ 𝐁𝐁 ∧ 𝐂𝐂 = 𝐀𝐀 ∧ (𝐁𝐁 ∧ 𝐂𝐂)
 𝐀𝐀 ∧ 𝐁𝐁 ∨ 𝐂𝐂 = 𝐀𝐀 ∧ 𝐁𝐁 ∨ 𝐀𝐀 ∧ 𝐂𝐂
 𝐀𝐀 ∨ 𝐁𝐁 ∧ 𝐂𝐂 = 𝐀𝐀 ∨ 𝐁𝐁 ∧ (𝐀𝐀 ∨ 𝐂𝐂)
 𝐀𝐀 ⊙ 𝐁𝐁 ⊙ 𝐂𝐂 = 𝐀𝐀 ⊙ (𝐁𝐁 ⊙ 𝐂𝐂)
30© S. Turaev, CSC 1700 Discrete Mathematics