Make five solved problems each using:
a.Golden ratio
b.Translating English sentence into mathematical sentences
D.Compound statements into symbolic form
E. Symbolic form into compound statement
F. Construction of truth table
G. Inverse converse and contrapositive
Sure, here are some solved problems for each of the topics you mentioned:
a. Golden Ratio:
1. Find the value of x that satisfies the equation (x/5) = (5/(x-5)) using the golden ratio.
Solution: We can use the golden ratio to solve this equation by setting x-5 equal to the golden ratio (1.618...). So we have x-5 = 1.618..., which gives us x = 6.618....
2. A rectangle has a length of 8 units and a width of 5 units. What is the ratio of the length to the width that approximates the golden ratio?
Solution: The golden ratio is approximately 1.618, so we can divide the length of the rectangle (8) by 1.618 to get the approximate value of the width. This gives us a width of approximately 4.94 units. Therefore, the ratio of length to width is approximately 8:4.94, which is close to the golden ratio.
b. Translating English Sentences into Mathematical Sentences:
1. The product of 5 and a number is 35. Write this sentence in the form of an equation.
Solution: Let x be the number. Then, the sentence can be written as 5x = 35.
2. Three more than twice a number is equal to 17. Write this sentence in the form of an equation.
Solution: Let x be the number. Then, the sentence can be written as 2x + 3 = 17.
D. Compound Statements into Symbolic Form:
1. If it rains, then the ground will get wet. Write this statement in symbolic form.
Solution: Let P stand for "It rains" and let Q stand for "The ground gets wet". Then, the statement can be written as P → Q.
2. John is a doctor and he is rich. Write this statement in symbolic form.
Solution: Let P stand for "John is a doctor" and let Q stand for "John is rich". Then, the statement can be written as P ∧ Q.
E. Symbolic Form into Compound Statement:
1. Write the statement ¬(P ∧ Q) in words.
Solution: The statement ¬(P ∧ Q) can be read as "not both P and Q".
2. Write the statement P ∨ (Q ∧ ¬R) in words.
Solution: The statement P ∨ (Q ∧ ¬R) can be read as "either P is true, or both Q and not R are true".
F. Construction of Truth Table:
1. Construct a truth table for the statement P → (Q ∧ R).
Solution:
| P | Q | R | Q ∧ R | P → (Q ∧ R) |
|---|---|---|-------|-------------|
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | F | F |
| T | F | F | F | F |
| F | T | T | T | T |
| F | T | F | F | T |
| F | F | T | F | T |
| F | F | F | F | T |
2. Construct a truth table for the statement (P ∧ Q) → (Q ∨ R).
Solution:
| P | Q | R | P ∧ Q | Q ∨ R | (P ∧ Q) → (Q ∨ R) |
|---|---|---|-------|-------|------------------|
| T | T | T | T | T | T |
| T | T | F | T | T | T |
| T | F | T | F | T | T |
| T | F | F | F | F | T |
| F | T | T | F | T | T |
| F | T | F | F | T | T |
| F | F | T | F | T | T |
| F | F | F | F | F | T |
G. Inverse, Converse, and Contrapositive:
1. If P then Q. Write the inverse of this statement.
Solution: The inverse of "If P then Q" is "If not P then not Q".
2. If P then Q. Write the contrapositive of this statement.
Solution: The contrapositive of "If P then Q" is "If not Q then not P".