Explanation

An order on a set "S" is typically denoted by the symbol "≤" (less than or equal to).

It representing a relation that establishes a sense of ordering between elements within the set. 

Explanation

A compound proposition that is always true, regardless of the truth values of its components, is called a tautology.

The statement "P ∨ ¬P" (P or not P) is a tautology because it is always true, whether P is true or false.

Explanation

To find the principal (P), use the formula:

Simple Interest (SI) = (P × Rate × Time) / 100

Rearrange the formula to solve for P:

P = (SI × 100) / (Rate × Time)

P = (84 × 100) / (4 × 2)

P = 1050

Explanation

1. Pipe A fills the tank in 15 minutes (rate = $\frac{1}{15}$ per minute).

2. Pipe B fills the tank in 20 minutes (rate = $\frac{1}{20}$ per minute).

3. Combined rate: $\frac{1}{15}+\frac{1}{20}=\frac{7}{60}$ per minute.

4. Time taken together: $\frac{1}{\frac{7}{60}}=\frac{60}{7}$ minutes.

Explanation

To find the cost price (CP), use the formula:

CP = Selling Price (SP) / (1 + Gain %)

CP = 48 / (1 + 0.20)

CP = 48 / 1.20

CP = 40

Explanation

• To convert a decimal to a percentage, multiply by 100:
  

$0.09\times 100=9\mathrm{\%}$

Explanation

• Calculate 3% of 20:

$3\mathrm{\%}\times 20=\frac{3}{100}\times 20=0.6$

Explanation

• Distribute the 2 across the terms inside the parentheses:
  

$2(x^2+y^2+xy)=2x^2+2y^2+2xy$

Explanation

729 = 27 × 27

√729 = √(27 × 27)

√729 = 27

Explanation

The sequence follows a pattern of adding 7:

102 + 7 = 109

109 + 7 = 116

116 + 7 = 123

123 + 7 = 130

130 + 7 = 137

This is an arithmetic sequence with a common difference of 7.