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.