Step-by-Step Solutions for Mathematical Problems

Set Difference Calculation

To find the set difference A - B, we identify all elements present in set A but not in set B.

Step-by-Step Subtraction

• Is 1 ∈ B? No. (Keep 1)
• Is 2 ∈ B? No. (Keep 2)
• Is 3 ∈ B? Yes. (Remove 3)
• Is 5 ∈ B? No. (Keep 5)
• Is 7 ∈ B? Yes. (Remove 7)
• Is 8 ∈ B? No. (Keep 8)

The remaining elements from set A are {1, 2, 5, 8}.

Symbolic Logic

In symbolic logic, the word "but" functions like "and," indicating that both conditions occur simultaneously. To write "He is rich but not generous" in symbolic form:

• p: "He is rich"
• q: "He is generous"
• ¬q: "He is not generous"
• ∧: The conjunction operator

Logic Symbol Reference

Logarithm Calculations

To find the value of log 360, we break 360 into factors of 2, 3, and 10.

Step 1: Factorize 360

We can write 360 as 2³ × 3² × 5 (or factors of 10).

Step 2: Apply Logarithm Properties

Using log(ab) = log a + log b and log(aⁿ) = n log a, we expand the expression.

Step 3: Substitute Values

Given log 10 = 1, we substitute these into the equation to find the final answer: log 360 = 2.5562.

Arithmetic Progression

If the terms (x + 1), 3x, and (4x + 2) are in an Arithmetic Progression (A.P.), the common difference between consecutive terms must be equal.

Step 1: Set up the Equation

The common difference (d) is calculated as: 3x - (x + 1) = (4x + 2) - 3x.

Step 2: Simplify and Solve for x

• Simplify: 2x - 1 = x + 2
• Isolate x: x = 3

Step 3: Verification

If x = 3, the terms are 4, 9, 14, which is an A.P. with a common difference of 5.

Matrix Operations

Determinant of Matrix A

Given matrix A, we expand along the first row to find the determinant |A| = 1.

Solving Matrix Equation AX = B

To find X, we use the formula X = A⁻¹B. After calculating the inverse and multiplying by B, we obtain the final matrix.

Logical Equivalence and Tautologies

Logical Equivalence

To show that p → q and its contrapositive ¬q → ¬p are logically equivalent, we construct a truth table. Since the final columns are identical, they are equivalent.

Proving a Tautology

The statement (p → q) ↔ (¬q → ¬p) is a tautology because it is true for every possible combination of truth values.

Survey Data Analysis

In a survey of 550 TV watchers, we use the inclusion-exclusion principle to determine viewership habits.

Key Findings

• Watch all three games: 20
• Watch exactly one game: 325

Arithmetic and Geometric Series

For an A.P. where the 7th term is 23 and the 12th term is 38, we solve the system of equations to find the first term and common difference. For the geometric series 2, -4, 8, ..., -4096, we identify the parameters and calculate the sum using the geometric series formula.