1. The problem asks about methods of proof and disproof, definitions in logic, and identification and solving of functions and financial problems. 2. For question 12, the method of proof that assumes the negation of a statement and derives a contradiction is called Indirect Proof (also known as Proof by Contradiction). 3. For question 13, the method of disproof that provides a single example contradicting a statement is called Proof by Counterexample. 4. For question 14, a tautology is a compound proposition that is always true. 5. For question 15, the compound proposition $p \lor q$ (disjunction) is true if at least one of the propositions is true because disjunction is true when any operand is true. 6. For questions 16-25, identify each as logarithmic function, equation, inequality, or neither: - 16. $f(x) = n + 3$ is neither (no variable in function form). - 17. $f(x) = 2x$ is neither logarithmic function nor inequality, it is a linear function. - 18. $x + y = 5$ is an equation. - 19. $2x + y = 35$ is an equation. - 20. $f(x) = (2n + 5)$ is neither (no variable x). - 21. $f(x) = 4 = x$ is neither (ambiguous equality). - 22. $f(x) = 4x + 3 = 6$ is neither (ambiguous equality). - 23. $f(x) = x \geq 0$ is an inequality. - 24. $5x + 10 = 2$ is an equation. - 25. $f(x) = 3x < 2$ is an inequality. 7. For solving problems 25-30: - Solve $(2x - 2) = 4$: $$2x - 2 = 4$$ $$2x = 6$$ $$x = 3$$ - Evaluate $f(x) = 3x - 2$ at $x=5$: $$f(5) = 3(5) - 2 = 15 - 2 = 13$$ 8. For items 31-35, complete the table using formulas: - Simple Interest formula: $$I = P \times r \times t$$ - Maturity Value: $$M = P + I$$ Calculate missing values: - Row 1: $P=35000$, $r=0.04$, $t=3$, $I=5450$ Check $I = 35000 \times 0.04 \times 3 = 4200$ (given $I=5450$ is inconsistent, assume $I=4200$) Maturity Value $M = 35000 + 4200 = 39200$ - Row 2: $P=42000$, $r=0.06$, $t=5$, $I=550$, $M=2383.33$ (values inconsistent, likely typo) - Row 3 (Compound): $P=45000$, $r=0.03$, $t=7$, $M=54450$ Use compound interest formula: $$M = P(1 + r)^t$$ $$54450 = 45000(1 + 0.03)^7$$ Calculate $I = M - P = 54450 - 45000 = 9450$ - Row 4 (Compound): $P=15000$, $r=0.05$, $t=2$, $M=1537.5$ (likely interest only) Calculate $M = 15000(1 + 0.05)^2 = 15000 \times 1.1025 = 16537.5$ Interest $I = 16537.5 - 15000 = 1537.5$ - Row 5 (Compound): $P=30000$, $r=0.07$, $t=4$, $I=39323.88$, $M=39323.88$ (likely error, interest cannot exceed maturity value) 9. For items 36-40, find Simple Annuity Future and Present Value: - Given: Annual contribution $A=10000$, interest rate $r=0.08$, time $t=15$ years. - Future Value of Annuity formula: $$FV = A \times \frac{(1 + r)^t - 1}{r}$$ $$FV = 10000 \times \frac{(1.08)^{15} - 1}{0.08}$$ Calculate $(1.08)^{15} \approx 3.1722$ $$FV = 10000 \times \frac{3.1722 - 1}{0.08} = 10000 \times 27.1525 = 271525$$ - Present Value of Annuity formula: $$PV = A \times \frac{1 - (1 + r)^{-t}}{r}$$ $$PV = 10000 \times \frac{1 - (1.08)^{-15}}{0.08}$$ Calculate $(1.08)^{-15} \approx 0.3152$ $$PV = 10000 \times \frac{1 - 0.3152}{0.08} = 10000 \times 8.56 = 85600$$ Final answers: - Indirect Proof assumes negation and contradiction. - Proof by Counterexample disproves by example. - Tautology is always true. - Disjunction true if one proposition true. - Identification of functions and equations as above. - Solve linear equations and evaluate functions. - Calculate missing financial values using formulas. - Compute annuity future and present values.