Question

It is given that
\(\times\) denotes greater than,
\(\phi\) denotes equal to,
\(<\) denotes not less than,
\(\perp\) denotes not equal to,
\(\Delta\) denotes less than,
\(+\) denotes not greater than.
Choose the correct statement from the following-
If a \(\times\) b \(\Delta\) c it follows that-

• A. a \(\phi\) c \(\Delta\) b
• B. b \(<\) a \(\times\) c
• C. a \(<\) b \(+\) c
• D. c \(+\) b \(<\) a

Hint:

Always simplify "not less than" (\(\not<\)) to "greater than or equal to" (\(\ge\)) and "not greater than" (\(\not>\)) to "less than or equal to" (\(\le\)) immediately. It avoids double-negative confusion during calculations.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to translate the coded mathematical symbols into standard inequality operators and evaluate which of the given options logically follows from the statement "a \(\times\) b \(\Delta\) c".

Step 2: Key Formula or Approach:
First, let us construct a translation key for all the symbols:
- \(\times \rightarrow >\) (greater than)
- \(\phi \rightarrow =\) (equal to)
- \(< \rightarrow \not<\) which is \(\ge\) (not less than)
- \(\perp \rightarrow \neq\) (not equal to)
- \(\Delta \rightarrow <\) (less than)
- \(+ \rightarrow \not>\) which is \(\le\) (not greater than)

Step 3: Detailed Explanation:
Translate the given expression "a \(\times\) b \(\Delta\) c":
- "a \(\times\) b" \(\rightarrow a > b\)
- "b \(\Delta\) c" \(\rightarrow b < c\)
So, the given condition is: \( a > b \) and \( b < c \).
Now let us translate and evaluate each option:
- Option (A): "a \(\phi\) c \(\Delta\) b"
- Translation: \( a = c \) and \( c < b \).
- Since we are given \( b < c \), \( c < b \) is false. Thus, Option (A) is incorrect.
- Option (B): "b \(<\) a \(\times\) c"
- Translation: \( b \ge a \) and \( a > c \).
- Since we are given \( a > b \), \( b \ge a \) is false. Thus, Option (B) is incorrect.
- Option (C): "a \(<\) b \(+\) c"
- Translation: \( a \ge b \) and \( b \le c \).
- We are given \( a > b \), which logically implies \( a \ge b \) is true.
- We are given \( b < c \), which logically implies \( b \le c \) is true.
- Since both parts of the inequality are true, Option (C) is correct.
- Option (D): "c \(+\) b \(<\) a"
- Translation: \( c \le b \) and \( b \ge a \).
- Since we are given \( b < c \), \( c \le b \) is false. Thus, Option (D) is incorrect.

Step 4: Final Answer:
(C) a \(<\) b \(+\) c