MATH 7 REVIEWER

1. MATH 7 REVIEWER
**Well-Defined Sets:**
- **Definition:** A set is well-defined if it is possible to determine whether an
object belongs to the set.
- **Example:** The set of all natural numbers less than 10, ( {1, 2, 3, 4, 5, 6, 7,
8, 9} ).
**Subsets:**
- **Definition:** A set ( A ) is a subset of set ( B ) if all elements of ( A) are also
elements of ( B ). Denoted as ( A subset B ).
- **Example:** If ( B = {1, 2, 3, 4, 5} ) and ( A = {2, 4} ), then ( A subseteq B ).
**Universal Sets:**
- **Definition:** The universal set ( U ) contains all objects under
consideration, usually represented by ( U ).
- **Example:** If considering natural numbers less than 10, ( U = {0, 1, 2, 3, 4,
5, 6, 7, 8, 9} ).
**Null Sets:**
- **Definition:** A null set (or empty set) has no elements, denoted by
(emptyset ) or ( {} ).
- **Example:** The set of natural numbers greater than 10 but less than 11,
( emptyset ).
**Cardinality of Sets:**
- **Definition:** The cardinality of a set is the number of elements in the set.
- **Example:** If ( A = {1, 2, 3} ), the cardinality of ( A ) is 3, denoted as ( |A| =
3 ).
The Real Number System and Its Subsets
**Natural Numbers (N):**
- **Definition:** Counting numbers starting from 1.
- **Example:** ( {1, 2, 3, …})
**Whole Numbers (W):**
- **Definition:** Natural numbers including 0.
- **Example:** ( {0, 1, 2, 3, …} )
**Integers (Z):**
- **Definition:** Whole numbers and their negative counterparts.
- **Example:** ( {…, -3, -2, -1, 0, 1, 2, 3, …} )
**Rational Numbers (Q):**
- **Definition:** Numbers that can be expressed as the ratio of two integers.
**Irrational Numbers:**
- **Definition:** Numbers that cannot be expressed as the ratio of two
integers. Their decimal expansions are non-repeating and non-terminating.

2. **Real Numbers (R):**
- **Definition:** All rational and irrational numbers.
- **Example:** ( {dots, -sqrt{3}, -1, 0, frac{1}{2}, pi, 3, sqrt{2}, dots} )
**Relationships within the Real Number System:**
- Natural Numbers ( subset ) Whole Numbers ( subset ) Integers ( subset
) Rational Numbers ( subset ) Real Numbers
- Irrational Numbers ( subset ) Real Numbers
Comparing, Ordering, and Representing Real Numbers on a Number Line
**Number Line Representation:**
- Plot rational and irrational numbers on a number line.
- Justify their placement by determining their value relative to other numbers.
**Example:**
- Place ( sqrt{2} approx 1.41 ), ( frac{3}{2} = 1.5 ), and ( pi approx 3.14 )
on a number line.
- ( sqrt{2} ) is between 1 and 2, closer to 1.5.
- ( frac{3}{2} ) is exactly 1.5.
- ( pi ) is between 3 and 4, closer to 3.
**Absolute Value:**
- **Definition:** The distance of a number from 0 on a number line, always
non-negative.
- **Example:** ( |3| = 3 ), ( |-3| = 3 )
**Distance Between Two Integers:**
- **Formula:** Distance between ( a ) and ( b ) is ( |a - b| ).
- **Example:** Distance between -2 and 5 is ( |-2 - 5| = |-7| = 7 ).
Properties of Operations on the Set of Integers
**Addition of Integers:**
1. **Adding Two Positive Integers:** The sum is always positive.
- Example: ( 3 + 5 = 8 )
2. **Adding Two Negative Integers:** The sum is always negative.
- Example: ( -4 + (-6) = -10 )
3. **Adding One Positive and One Negative Integer:** The sum depends on
the larger absolute value.
- Example: ( 7 + (-3) = 4 ), ( -7 + 3 = -4 )
4. **Real-World Situations:**
- Gaining or losing money: ( $50 + (-$20) = $30 )
- Temperature changes: ( 10^circ + (-15^circ) = -5^circ )
**Subtraction of Integers:**
1. **Using a Number Line:**
- Example: ( 5 - 3 ) is moving 3 units left from 5.
2. **Subtracting a Negative Integer:** Equivalent to adding its positive
counterpart.

3. - Example: ( 5 - (-3) = 5 + 3 = 8 )
3. **Real-World Problems:**
- Temperature changes: ( 20^circ - 5^circ = 15^circ )
- Financial balances: ( $50 - (-$20) = $70 )
**Multiplication and Division of Integers:**
1. **Same Sign:** The product/quotient is positive.
- Example: ( 3 times 4 = 12 ), ( -3 times (-4) = 12 )
2. **Different Signs:** The product/quotient is negative.
- Example: ( -3 times 4 = -12 ), ( 12 div (-4) = -3 )
3. **Real-World Contexts:**
- Profit and loss: ( 4 times (-5) = -20 ) (loss)
- Elevation changes: ( -2 times 3 = -6 ) (descending)
**Order of Operations (GEMDAS):**
1. **Grouping Symbols:** Parentheses, brackets, etc.
2. **Exponents:** Powers and roots.
3. **Multiplication/Division:** From left to right.
4. **Addition/Subtraction:** From left to right.
**Simplifying Expressions:**
- **Example:** ( 3 + (2 times 3^2 - 4) )
- Calculate inside parentheses first: ( 2 times 3^2 - 4 = 2 times 9 - 4 = 18 -
4 = 14 )
- Then add: ( 3 + 14 = 17 )