Study Notes/Grade 7 Math/Sets & Integers

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Lesson 1 • 40 min read

Sets & Integers

In This Lesson

→ Types of Sets → Set Operations → Number Line → Integer Operation Rules → Order of Operations (PEMDAS)

1Types of Sets

Finite Set

A set with a countable number of elements. You can list all members.

Example: A = {1, 2, 3, 4, 5}

Infinite Set

A set with unlimited elements that go on forever.

Example: N = {1, 2, 3, 4, ...} (natural numbers)

Empty Set (Null Set)

A set with no elements. Denoted as ∅ or { }.

Example: Set of months with 32 days = ∅

Subsets

Set A is a subset of B if every element of A is also in B. Written as A ⊆ B.

If A = {1, 2} and B = {1, 2, 3, 4}, then A ⊆ B

Universal Set

The set that contains all elements under consideration for a particular problem. Denoted as U.

2Set Operations

Union (∪)

All elements in either set (combine both)

A = {1, 2, 3}

B = {3, 4, 5}

A ∪ B = {1, 2, 3, 4, 5}

Intersection (∩)

Elements common to both sets

A = {1, 2, 3}

B = {3, 4, 5}

A ∩ B = {3}

Difference (A - B)

Elements in A but not in B

A = {1, 2, 3}

B = {3, 4, 5}

A - B = {1, 2}

Complement (A')

Elements not in set A (from Universal set)

U = {1, 2, 3, 4, 5}

A = {1, 2, 3}

A' = {4, 5}

3The Number Line

... -3, -2, -1, 0, +1, +2, +3 ...

← Negative Numbers | Zero | Positive Numbers →

Negative Integers

Numbers less than zero. Located to the left of zero on the number line.

Positive Integers

Numbers greater than zero. Located to the right of zero on the number line.

Absolute Value |x|

The distance from zero on the number line. Always positive or zero.

|5| = 5, |-5| = 5, |0| = 0

4Integer Operation Rules

Addition Rules

Same Signs: Add and keep the sign

(+5) + (+3) = +8

(-5) + (-3) = -8

Different Signs: Subtract and take the sign of the larger absolute value

(+5) + (-3) = +2 (|5| > |3|, so positive)

(-5) + (+3) = -2 (|-5| > |3|, so negative)

Multiplication/Division Rules

Same Signs = POSITIVE

(+) × (+) = (+)

(-) × (-) = (+)

Different Signs = NEGATIVE

(+) × (-) = (-)

(-) × (+) = (-)

The same rules apply for division!

Quick Memory Trick

"Same sign = Positive, Different sign = Negative"

Think: Friends (+,+) or (-,-) agree. Enemies (+,-) disagree.

5Order of Operations (PEMDAS/GEMDAS)

PEMDAS Steps

Parentheses / Grouping Symbols

( ), [ ], { }, fraction bars

Exponents

Powers like 2³, 5²

Multiplication and Division

Left to right (equal priority)

Addition and Subtraction

Left to right (equal priority)

Example Problem

Solve: 3 + 4 × 2² - (6 ÷ 2)

Step 1 (P): 3 + 4 × 2² - 3

Step 2 (E): 3 + 4 × 4 - 3

Step 3 (M): 3 + 16 - 3

Step 4 (AS): 19 - 3 = 16

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