Math 101: Set Equality and Universal Set

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Questions and Answers

How many soldiers were shot in all three parts of the body?

4 (correct)

What is the defining characteristic of an ordered pair?

The elements must not belong to the same set.

The elements must be integers.

The order of the elements matters. (correct)

The elements can be swapped without changing the pair.

Which statement best describes the Cartesian product of sets A and B?

It includes only pairs where elements are the same from both sets.

It contains only the elements of A.

It is the set of all ordered pairs where the first element is from A and the second is from B. (correct)

It consists of all combinations of elements from A and B as unordered pairs.

Which of the following relations could be categorized as a binary relation?

All of the above Signup and view all the answers

What is true about the set {x, y}?

It is an unordered pair of elements. Signup and view all the answers

In the ordered pair (2, 3), what is the significance of the numbers’ positions?

The first number represents input while the second is output. Signup and view all the answers

If R is a relation defined on set A, what does the ordered pair (x, y) signify?

R exists for that specific ordered pair (x, y). Signup and view all the answers

What type of statements can a relation represent?

True or false propositions about the relationship between elements. Signup and view all the answers

Which statement correctly describes when two sets A and B are equal?

A is a subset of B and B is a subset of A. Signup and view all the answers

What symbol is commonly used to denote the universal set?

E Signup and view all the answers

What is the result of the union of the sets A = {1, 2, 3} and B = {3, 4, 5}?

{1, 2, 3, 4, 5} Signup and view all the answers

If A = {1, 2, 3} and B = {2, 3, 4}, what is A intersection B?

{2, 3} Signup and view all the answers

Which of the following correctly describes the complement of a set A?

It consists of elements not in A but in the universal set. Signup and view all the answers

If a set A has 3 elements, how many subsets does the power set of A contain?

8 Signup and view all the answers

Which property of union and intersection states that A ∩ (B ∪ C) equals (A ∩ B) ∪ (A ∩ C)?

Distributive Property of Intersection over Union Signup and view all the answers

What does the expression A ∪ A represent?

The set A itself. Signup and view all the answers

What is the expanded form of sin3A based on the given formulations?

3sinA - 4sin^3A Signup and view all the answers

What does tan3A equate to in terms of tanA according to the formulas provided?

(3tanA - tan^3A) / (1 - 3tan^2A) Signup and view all the answers

If cosA = 0.5, what is the value of sin2A calculated using the corresponding identity?

0.25 Signup and view all the answers

Which of the following correctly represents the half angle formula for sinA?

sinA = √((1 - cosA)/2) Signup and view all the answers

Using the double angle formula, which expression is equivalent to cos2A?

cos^2A - sin^2A Signup and view all the answers

What is the formula for sin(A + B)?

sinAcosB + cosAsinB Signup and view all the answers

What does the equation sin(A + B) + sin(A − B) simplify to?

2sinAcosB Signup and view all the answers

When applying the angle sum and difference identities to cos(A + B) and cos(A − B), which result is obtained from their addition?

2cosAcosB Signup and view all the answers

What is the correct formula for cos(A − B)?

cosAcosB - sinAsinB Signup and view all the answers

Substituting A + B = α and A − B = β, what value represents A?

$rac{α + β}{2}$ Signup and view all the answers

What is the result of cos(A + B) − cos(A − B)?

−2sinAsinB Signup and view all the answers

Which of the following identities represents the relationship for sin(A + B)?

sin(A + B) = sinAcosB + cosAsinB Signup and view all the answers

What is the modulus of the complex number $z = -4 + 2i$?

$\sqrt{20}$ Signup and view all the answers

Which of the following properties of complex numbers is represented by the inequality $|z_1 - z_2| \geq |z_1| - |z_2|$?

Reverse triangle inequality Signup and view all the answers

In the complex or Argand plane, what does the point $(x, y)$ represent?

A complex number $z = x + iy$ Signup and view all the answers

What is the expression for the modulus of the complex number $z^k$ for $k \in \mathbb{N}$?

$|z|^k$ Signup and view all the answers

Which graphical representation technique is used for adding two or more complex numbers in the Argand plane?

Parallelogram rule Signup and view all the answers

What is the modulus of the complex number represented by the ordered pair $(3, -2)$?

$\sqrt{13}$ Signup and view all the answers

If $z_1 = 2 + i$ and $z_2 = 3 - 2i$, what is the modulus of $3z_1 - 4z_2$?

$\sqrt{34}$ Signup and view all the answers

Which equation correctly expresses the distance from the origin for a complex number $z = x + iy$?

$|z| = \sqrt{x^2 + y^2}$ Signup and view all the answers

Study Notes

Equality of a Set

Two sets A and B are equal if A ⊆ B and B ⊆ A.
Example: If X = {1, 2, 3} and Y = {3, 2, 1}, then X = Y.

Universal Set

The universal set is the total collection of elements in a given context.
Denoted by the symbol µ or E.

Union of a Set

The union of sets A and B, written as A ∪ B, includes elements from A, B, or both.
Example: A = {1, 2, 3}, B = {2, 3, 4} leads to A ∪ B = {1, 2, 3, 4}.

Intersection of a Set

The intersection of sets A and B contains common elements from both, denoted as A ∩ B.
Example: A = {1, 2, 3, 4}, B = {2, 4, 5} results in A ∩ B = {2, 4}.

Complement of a Set

The complement of a set A includes elements not in A but in the universal set.
Denoted as A' or Ac.

Power Set

The power set of A consists of all possible subsets of A.
If A has n elements, the number of subsets is 2^n.
Example: For A = {1, 2, 3}, P(A) has 2^3 = 8 subsets.

Properties of Union and Intersection

A ∪ A = A; A ∩ A = A.
A ∪ B = B ∪ A; A ∩ B = B ∩ A (commutative property).
A ∪ (B ∪ C) = (A ∪ B) ∪ C; A ∩ (B ∩ C) = (A ∩ B) ∩ C (associative property).
A ⊆ (A ∪ B) and B ⊆ (A ∪ B).
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (distributive property).
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Elements of Relations and Functions

An unordered pair {x, y} denotes subsets of set X; order does not matter.
An ordered pair (x, y) gives preference to the order, where (x, y) ≠ (y, x) if x ≠ y.
The Cartesian product X × Y is the set of all ordered pairs (x, y) such that x ∈ X and y ∈ Y.

Relations

A relation R exists on set A if xRy holds true for any ordered pair (x, y) ∈ A.
Relations can express statements like “is greater than” or “is a sibling of”.

Complex or Argand Plane

Complex numbers can be represented as an ordered pair (x, y) corresponding to coordinates on a plane.
The distance from the origin relates to |z| = √(x^2 + y^2).
The x-axis is the real axis, and the y-axis is the imaginary axis.

Addition of Complex Numbers

Use the parallelogram rule for addition in the Argand plane, similar to vectors.

Factor Formulae

Sin and cos functions can be expressed in terms of sums and differences:
- sin(A + B) = sinAcosB + cosAsinB
- cos(A + B) = cosAcosB − sinAsinB
These can be manipulated to derive relationships involving sin(2A), cos(2A), etc.

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Description

This quiz covers the concepts of set equality and universal sets as discussed in Math 101: General Mathematics I. You will learn how two sets can be equal and the significance of the universal set in mathematical contexts, with examples provided to illustrate these concepts.