A car is traveling at a constant speed of 30 mi/hr.
30πππππ 1βπ = 90πππππ
3βπ =15πππππ
(12)βπ Note since the speed is constant the proportion is valid
30πππππ
1/3 of a tank contains 10 gallons. The fraction of the tank that is filled is proportional to the gallons it holds.
(13) π‘πππ
After World War II, 1939 β 1945, through the Marshall Plan, which involved more than 100 million people and about 30 countries, the US helped rebuild Europe. Many Germans were misled by Adolf Hitler, but Germany and Japan were defeated. It was time to look forward and rebuild relationships. Germany and Japan are presently some of the US best allies. We need to study and understand history. By the way, we all make mistakes. Keep this in mind when judging others.
Presently Germany is trying to help some African nations. Perhaps this could be an incentive for many Africans to stay home and help rebuild their nation. Presently, they are migrating to Europe. Corruption seems to be what holds back some of these African nations. Sometimes the source of corruption may be a foreign company which exploits cheap slavery among children. Helping is a good idea but knowing how to help is essential.
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Lesson 35 Logic: In Mathematics Statements or Propositions are either True or False In mathematics and in logic, a statement must be either true or false. For example, the statement it is raining, and it is not raining, canβt be true and false. This is not a statement. Another example, the statement x = 3 is a solution and x = 3 is not a solution, has no acceptance in mathematics.
In mathematics, the symbols p and q are frequently used to represent statements. Let me illustrate, p may represent the statement it is raining and q the statement the grass is wet.
A statement derived from these two is: it is raining and the grass is wet, p and q.
The negation of a statement
It is not raining is the negation of the statement it is raining. The negation of p is ~ p.
Q1 If the statement p, it is raining is true, what about the statement ~ (~ p)?
Answer if p is T, ~ p, its negative, is F, and its negative ~ (~ p) is T.
The negative of it is raining is it is not raining. and its negative is no, it is not raining. OK The addition (pβ§q) and subtraction (p β¨ q) of statements p, q
Examples
It is raining and I am going out to play, is a true statement if both p and q were true.
Q2 Is the statement x = 5 is a root and x = 5 is not a root a valid mathematical statement?
Answer it is not a mathematical statement. A statement and its negative canβt be both true or false.
Q3 Is the statement x = 2 is a root or x = 2 is not a root a valid mathematical statement?
Answer the statement is not a valid one. . The statement p or q is T if either are true or if both are true.
Conditional statement p β q, is read p implies q or if p then q Example of a conditional statement: βIf it is raining, then the grass is wet.β
A conditional statement has two parts, a hypothesis, p, and a conclusion, q.
A conditional statement is called and if then statement and is represented by the symbol p β q. It is also read βp implieq.β
Let me illustrate this type of statement with an example.
If you do this or do that you will be successful.
Suppose you were successful, but you forgot what you did. Shouldnβt the statement be T?
Consider the statement if it is raining then the grass will get wet.
Next morning you find the grass wet, immediately you infer it rained. But wait, the neighborβs grass on one side is wet but not the grass on the other side of your house. One neighbor used a sprinkler.
Q4 Is the statement if it is raining then the grass will get wet T/F?
The conditional statement p β q is T if q is T, regardless if p is true or false.
Let me illustrate with another example that may confuse you more!
Relationship between p β q and ~ p β ~ p Example
If you use road A or road B it will take you to point C Q5 But if you are not at point C, what is your conclusion?
Answer that you did not take road A or road B,
Q7 If you were not successful, what is your conclusion?
You failed to do A and B.
To reach point C you must use road A or road B (the roads are parallel to each other).
Example 2
If x = 3 than x2 = 9 is T. If x = β 3 then x2 = 9 is T.
Q1 The statement p β q is T regardless if p is T or F Answer x2 equals 9 if either x = 3 or x = β 3.
The statement If x2 β 9 then x β 3 and x β β 3.
Conditional statement
The validity of (p β¨ q) β t is identical to that of ~ t β (~ p β§ ~ q).
The validity of (p β§ q) β t is identical to that of ~ t β (~ p β¨ ~ q).
The conditional statement (p β¨ q) β t takes into consideration that there may be more than one hypothesis, in this case two. p and q.
1. If the statement βIf it rains the grass is wetβ is true, which of the following statements is also true?
Being part of a community
Pedro is a fisherman; every week he goes fishing with two other fishermen. They return on Saturday morning to sell their catch to the people in their community. It is an occasion of great celebration and many participate; Jose and Pedro sell empanadas and drinks; a young group of musicians makes the occasion a happier one. The local priest is present to thank God. However, sometimes they donβt catch very much. People understand that.
One week, after having a good catch, just when they are ready to return home, a large foreign boat approaches them and offers to buy all they had caught at a good price; much higher than the people in his community can afford. Pedro must decide, make lots of money or support his community. Will he take the much-needed food away from the table of people and friends who throughout the years have relied on him and helped him?
Math riddle
Consider the statement, if it rains the grass is wet. But wait, the grass is wet, yet it did not rain. Does it make sense that the statement if it rains the grass is wet be true even though it did not rain?
In your hypothesis you probably forgot or did not take into consideration other possible causes.
Slope of a line
With $12 you purchase x lb. of item A at $2/lb. and y lb. of item B at $4/lb. if you only bye item A, x = 6 lb. and y = 0 lb. but if you only bye item B, x = 0 lb. and y = 3lb. By giving up 6 lb. of A, you purchase 3 lb. of B. from (6, 0) to (0, 3), run = 6, rise = β 3, so the slope of 2x + 4y = 12 is β 1/2.
Q1 Why canβt both x and y increase?
Q2 When [2, 4], a vector, is rotated 900 CCW, it becomes [β 4, 2], a vector parallel to the line 2x + 4y = 12. So, [2, 4] is perpendicular to the line.
Our greatest need
When our neighbor needs us, are we available?
Chaos will overtake us when we fail to love God and our neighbor, or we fail to give God and our neighbors priority in our lives.
The diagram helps to answer these questions.
a. If the grass is wet then it rained. T/F
b. If it has not rained then the grass is not wet. T/F c. If the grass is not wet then it has not rained. T F
Answer a. F a point may be inside large circle but outside small circle rain b. F not rained is a point outside the small circle, grass may be wet or dry c. T a point on the region dry grass cannot belong to the circle rain.
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Lesson 36 Transition from Arithmetic to Algebra
Distributive property in Arithmetic
When calculating (2)(3 + 4), is the addition done first? It does not matter; see the figure below.
2h
2(3) 2(4)
Four terms, four rectangles Two terms, two rectangles One term, one rectangle The sum of the areas of the four rectangles
(2)(4) + (2)(5) + (3)(4) + (3)(5) = 2(4 + 5) + 3(4 + 5) = (4 + 5)(2 + 3) 1. How many terms does 2(4 + 5) + 3(4 + 5) have, and what is the common factor?
Answer 1. 2. 2(4 + 5) + 3(4 + 5) = (4 + 5)(2 + 3). (4 + 5) is the common factor or common base.
2. To evaluate the expression (4 + 5)(2 + 3), do we first add or do we first multiply?
It doesnβt matter. The area of the large rectangle, 5(9) = 45 is the same as the area of the four rectangles, 2(4) + 2(5) + 3(4) + 3(5) = 8 + 10 + 12 + 15 = 45
2h is the common height of the three rectangles.
2h(3 + 4)b is the area of the large rectangle.
(2h)(3b) and (2h)(4b) are the areas of the two smaller rectangles.
a is the common height of the three rectangles.
a(b + c) is the area of the large rectangle.
a(b) and a(c) are the areas of the two smaller rectangles.
The area of the two lower rectangles is
2h(4) + 2h(5) = 2h(4 + 5) = 2h(9) = area of the 2 by 9 rectangle.
The area of the two top rectangles is
3h(4) + 3h(5) = 3h(4 + 5) = 3h(9) = area of the 3 by 9 rectangle.
2(3 + 4) = (2)(3) + (2)(4) helps the transition from Arithmetic to Algebra