Fundamental Concepts of Natural Numbers

Questions and Answers

Which law states that a + b = z?

Commutative law for addition

Distributive law

Associative law for addition

Law of closure for addition (correct)

The commutative law for multiplication states that a x b = b x a.

True

What does the associative law for addition state?

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

Which of the following expressions represents the distributive law?

a(b + c) = ab + ac Signup and view all the answers

What is the reflexive property?

a = a Signup and view all the answers

What does transitive property state?

If a = b and b = c, then a = c. Signup and view all the answers

What does the inequality symbol 'a > b' mean?

a is greater than b Signup and view all the answers

If x < 1 or x > 5, then x can be a value between 1 and 5.

False Signup and view all the answers

What is the theorem related to adding the same number to both sides of an inequality?

If a > b, then a + c > b + c. Signup and view all the answers

What can be concluded from multiplying both sides of an inequality by the same positive number?

The sense of the inequality does not change. Signup and view all the answers

Study Notes

Basic Laws of Natural Numbers

Law of Closure for Addition: If a and b are natural numbers, then their sum a + b is also a natural number. Example: 2 + 5 = 7.
Commutative Law for Addition: The order of addition does not affect the sum, a + b = b + a. Example: 5 + 7 = 7 + 5 = 12.
Associative Law for Addition: Grouping does not affect the sum, a + (b + c) = (a + b) + c. Example: 12 + (13 + 22) = (12 + 13) + 22 = 47.
Law of Closure for Multiplication: If a and b are natural numbers, then their product a × b is also a natural number. Example: 2 × 4 = 8.
Commutative Law for Multiplication: The order of multiplication does not affect the product, a × b = b × a. Example: 12 × -3 = -3 × 12 = -36.
Associative Law for Multiplication: Grouping does not affect the product, a(b × c) = (a × b) × c. Example: 6 × (-4 × -3) = (6 × -4) × -3 = 72.
Distributive Law: This law links addition and multiplication, a(b + c) = ab + ac. Example: 3(5 + 8) = 15 + 24 = 39; 3(5 - 8) = 15 - 24 = -9.

Basic Laws of Equality

Reflexive Property: Any quantity is equal to itself, a = a.
Symmetric Property: If a = b, then b = a.
Transitive Property: If a = b and b = c, then a = c; things equal to the same thing are equal to each other.
Addition of Equals: If a = b and c = d, then a + c = b + d.
Multiplication of Equals: If a = b and c = d, then ac = bd.

Inequalities

Definition: An inequality expresses a relationship between two quantities, indicating one is greater or less than the other using symbols like >, <, ≥, and ≤.
Compound Inequalities: Can indicate a range of values using "and" (e.g., 1 < x ≤ 5 means x is between 1 and 5) or "or" (e.g., x < 1 or x > 5 means x is outside the interval (1, 5)).

Theorems of Inequalities

Theorem 1: Adding the same number to both sides of an inequality will not change the sense of the inequality. (If a > b, then a + c > b + c).
Theorem 2: Multiplying both sides of an inequality by a positive number preserves the inequality. (If a > b and c > 0, then ac > bc). Example: For 8 > -5, multiplying both sides by 3 gives 24 > -15.
Division of Inequalities: Similar to multiplication, dividing by a positive number maintains the inequality's sense. (If a > b and c > 0, then a/c > b/c).

Description

This quiz covers the basic laws of natural numbers, focusing on the laws of closure, commutative, and associative properties for addition. It provides examples to illustrate each law, helping students to grasp these fundamental concepts. Perfect for anyone looking to solidify their understanding of basic arithmetic operations.