Mathematics 3rd Quarter - Measurement, Units, Sets - PDF

**Mathematics 3rd Quarter** I. A. B. (the human body was the first ruler- dangkal, dipa, hakbang, talamapakan, dakot) II. C. The **U.S. Customary or English System of Measurement** is based on the British Imperial System. Its units evolved from nonstandard units of measure in medieval ti...

**Mathematics 3rd Quarter** I. A. B. (the human body was the first ruler- dangkal, dipa, hakbang, talamapakan, dakot) II. C. The **U.S. Customary or English System of Measurement** is based on the British Imperial System. Its units evolved from nonstandard units of measure in medieval times like the foot, yard, and inch.  D. The **Metric System** is based on decimals just like our numeration and monetary system. It uses Greek and Latin prefixes to make conversion from one unit to another easier.  E.  III. **Units of Measurement** **1. Length** **2. Mass/Weight** **3. Time** **4. Temperature** **5. Area** **6. Volume and Capacity** **7. Speed** 1.   2. - -   3.  4. 5. - -  6. -  **PYRAMID** A pyramid is a three- dimensional solid with a polygonal base and triangular faces that converge to a single point called the apex. +-----------------------------------+-----------------------------------+ |  | | +===================================+===================================+ | **Rectangular Pyramid** | **Triangular Pyramid** | | | | | A rectangular pyramid is a type | A triangular pyramid is a | | of pyramid where the base is a | three-dimensional figure with a | | rectangle. It has four triangular | triangular base and three | | faces that meet at the same apex, | congruent triangular faces. Each | | and each triangular face is | triangular face intersects the | | connected to one side of the | base at the same angle. | | rectangular base. | | +-----------------------------------+-----------------------------------+ | | | +-----------------------------------+-----------------------------------+ |  | | +-----------------------------------+-----------------------------------+  7. Kph or km/h means "kilometer per hour" m/s means "meter per second" **IV. THE LANGUAGE OF SETS** **SET** \- is a group or collection of objects. **ELEMENT** \- each object/member in a set. **CARDINALITY** \- the number of elements in a set. **FINITE SETS** \- a set with countable numbers of elements. **INFINITE SETS** \- a set with an infinite number of elements. T**HREE (3) WAYS IN NAMING A SET** **1. List or Roster Method** -is a set that can be described by listing all the elements of a set within a pair of braces. {} For example: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} **2. Descriptive Method** \- is a set that can be described by writing a description of its elements. For example: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Descriptive Method: A = set of counting numbers from 1 to 10. **3. Set Builder Notation** \- is a set that can be described using variables. For example: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Set-Builder Notation: A = {x\|x is a counting number from 1 to 10} Read as "A is the set of all x such that x is a counting number from 1 to 10" +-----------------------------------------------------------------------+ | Examples | | | | Roster Method | | | | 1. | | | | | | | | 2. | +=======================================================================+ | Descriptive Method | | | | 1. | | | | | | | | 2. | +-----------------------------------------------------------------------+ | Set-builder Notation | | | | 1. | | | | | | | | 2. | | | |  | +-----------------------------------------------------------------------+ |  | +-----------------------------------------------------------------------+ - - - - -  **V. RELATIONS ON SETS** EQUAL SETS \- If two sets have the same elements in them. Ex. {1, 9, 7, 5} and {7, 1, 5, 9} EQUIVALENT SETS \- if two sets have the same number of elements, though the elements are different. Ex. {1, 2, 3, 4, 5, \..., 11} and {2, 4, 6, \..., 22} {5, 10, 15, 20, 25} and {5, 6, 7, 8, 9} A = {numbers less than 5} A = {1, 2, 3, 4} B = {even numbers less than 6} B = {2, 4} C = {prime numbers} C = {2, 3, 5, 7} D = {numbers from 1 to 4} D = {1, 2, 3, 4} **Universal Sets and Subsets** **UNIVERSAL SET** -- contains **all** the **elements** being considered in a given situation. Ex. U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} **SUBSETS** -- A set 'A' is said to be a subset of B if every element of A is also an element of B, denoted as A ⊆ B. Even the null set is considered to be the subset of another set. Ex. A = {1, 2, 3, 4} B = {2, 4} C = {2, 3, 5, 7} D = {1, 2, 3, 4} Example: A = {1,2,3} Then {1,2} ⊆ A. Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}. Note: The set is also a subset of itself. If A is not a subset of B, then it is denoted as A /⊆ B. **PROPER SUBSETS** -- If A B and A ≠ B, then A is called the proper subset of B and it can be written as A⊂B. Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B = {2,5,7} But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.   **VI. Operations on Sets** **INTERSECTION \|A ∩ B\|** The INTERSECTION of two sets, written by **A ∩ B** is the set containing the elements that are **common** to **both sets**.  **DISJOINT SETS (A ∩ B) = {}** When A and B have **no common elements**, we say that A and B are DISJOINT SETS. **UNION OF SETS \|A Ս B\|** The UNION of two sets, written by A Ս B is the **sets of all elements that are in A or in B** , or in both A and B.  **DIFFERENCE OF TWO SETS \|A-B or B-A\|** If A and B are two sets, then their DIFFERENCE is given by **A-B or B-A**. **COMPLEMENT OF A SET** The **COMPLEMENT** of a set **A** is written as **A\'**. Whereas A\' is a set of **elements** that are **not** in **A**.