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22. In the following question, the symbols +,,,, and $ are used with the following meanings illustrated.

X+Y' means 'X is neither less than nor equal to Y.

'X Y' means 'X is either greater than or equal to Y.

X-Y means 'X is neither greater than nor smaller than Y.

XSY means X is smaller than v

"X/Y' means 'X is either less than or equal to Y

In the following question assuming the given statements to be true, find out which of the three conclusions 1 II and III given below the is/are definitely tree and mark your answer accordingly.

Statements:

M+J

J/Y

Y$K

IV. K-N

Conclusions:

K*j
J+n
K$M

Let’s break down the meanings of the symbols and analyze the statements and conclusions:

### Symbols:
- **X+Y** means "X is neither less than nor equal to Y," i.e., **X > Y**.
- **X,Y** means "X is either greater than or equal to Y," i.e., **X ≥ Y**.
- **X-Y** means "X is neither greater than nor smaller than Y," i.e., **X = Y**.
- **X$Y** means "X is smaller than Y," i.e., **X < Y**.
- **X/Y** means "X is either less than or equal to Y," i.e., **X ≤ Y**.

### Given Statements:
1. **M+J**: This means M > J.
2. **J/Y**: This means J ≤ Y.
3. **Y$K**: This means Y < K.
4. **K-N**: This means K = N.

### Conclusions:
1. **K+J**: Is K greater than J?
2. **J-N**: Is J equal to N?
3. **K$M**: Is K smaller than M?

### Step-by-Step Analysis:

1. **M > J** from statement 1.
2. **J ≤ Y** from statement 2.
3. **Y < K** from statement 3.
4. **K = N** from statement 4.

#### Conclusion 1: **K+J (K > J)**
- From statements 2 and 3, we know that **J ≤ Y** and **Y < K**.
- Therefore, **K > J** is true.

#### Conclusion 2: **J-N (J = N)**
- From statement 4, we know **K = N**, and from statement 3, **Y < K**.
- From statement 2, **J ≤ Y**, so **J < K** (as **Y < K**), meaning **J < N**.
- Therefore, **J ≠ N**, making this conclusion false.

#### Conclusion 3: **K$M (K < M)**
- From statement 1, **M > J**, and we have already proven that **K > J** in conclusion 1.
- Therefore, **M > K**, which means **K < M** is true.

### Final Answer:
- Conclusion **I (K > J)** is **true**.
- Conclusion **II (J = N)** is **false**.
- Conclusion **III (K < M)** is **true**.

Thus, the correct conclusions are **I and III**.

Would you like more details on any step or further clarification?

### 5 Related Questions:
1. What happens if we reverse the symbols, such as **M$J**?
2. How do we interpret the statements when both **M+J** and **J+M** are true?
3. Can conclusions **I, II, and III** change if one of the statements is modified?
4. What other logical connectives or symbols could be used in similar problems?
5. How can we translate these symbols into set theory notations?

### Tip:
In problems with symbolic logic, carefully translate each symbol into a known inequality or equality to avoid confusion.

In the following question, the symbols +,,,, and $ are used with the following meanings illustrated. X+Y means 'X is neither less than nor equal to Y.' X,Y means 'X is either greater than or equal to Y.' X-Y means 'X is neither greater than nor smaller than Y.' X$Y means X is smaller than Y. X/Y means 'X is either less than or equal to Y.' Assuming the given statements to be true, which of the conclusions are definitely true: Statements: M+J J/Y Y$K K-N Conclusions: K+J J-N K$M?

Analyze a problem involving symbolic logic and inequalities to determine the truth of various conclusions. This question explores symbolic relationships between variables like greater than, smaller than, and equality. Suitable for students in Grades 10-12.

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