MTH202 Solution by Muhammad Umer Azam

Q1. Step 1: Symbolize the propositions

1. "Alice is a manager"

Let M represent " Alice is a manager."

2. "Bob is a supervisor"

Let S represent " Bob is a supervisor."

Step 2: Translate the premises into logical statements

1. Premise 1: "If Alice is a manager, then Bob is not a supervisor."

This is a conditional statement: M → ¬S (If Alice is a manager, then Bob is not a supervisor.)

2. Premise 2: "If Bob is a supervisor, then Alice is not a manager."

Another conditional statement: S → ¬M (If Bob is a supervisor, then Alice is not a manager.)

Step 3: Translate the conclusion

The conclusion is:

"Alice is not a manager, and Bob is not a supervisor."

This is a conjunction: ¬M ∧ ¬S

Step 4: Write the argument formally

The argument can now be written as:

1. M → ¬S (Premise 1)
2. S → ¬M (Premise 2)
3. ¬M ∧ ¬S (Conclusion)

We need to check whether the conclusion follows logically from the premises.

Step 5: Construct a truth table

We will list all possible truth values for M (Alice being a manager) and S (Bob being a supervisor) and evaluate the premises and the conclusion. M S ¬S M→¬S S→¬M ¬M ¬S ¬M∧¬S T T F F F F F F T F T T T F T F F T F T T T F F F F T T T T T T

Step 6: Evaluate the truth table

Row 3: In this case, M → ¬S and S → ¬M are both true. The conclusion ¬M ∧ ¬S is also true in this row. So, the argument holds true in this case. Other rows: In the other rows, at least one of the premises is false. For example: o Row 1: Both premises are false, so we don't need to check the conclusion. o Row 2: The first premise is true, but the second one is false. o Row 4: The second premise is true, but the first one is false.

Step 7: Conclusion

The argument is valid because in the case where both premises are true (row 3), the conclusion is also true. Therefore, the argument is logically valid based on the truth table.

Q2. To prove A∖B = A ∩ Bc

1. Show A∖B ⊆ A ∩ Bc:

If x ∈ A∖B, then x ∈ A and x ∉ B

So, x ∈ Bc

Thus, x ∈ A ∩ Bc

2. Show A ∩ Bc ⊆ A∖B:

If x ∈ A ∩ Bc , then x ∈ A and x ∈ Bc

So, x ∉ B

Thus, x ∈ A∖B

Conclusion: A∖ B= A ∩ Bc

Q3. Step 1: Set up the Venn diagram

Given:

o ∣C∩D∣=30 o ∣D∩M∣=25 o ∣C∩M∣=20 o ∣C∩D∩M∣=10 o No one attended only Machine Learning.

Breakdown:

o 20 attended only Coding and Data Science. o 15 attended only Data Science and Machine Learning. o 10 attended only Coding and Machine Learning. o 15 attended only Coding. o 10 attended only Data Science.

Total students:

15 + 10 + 20 + 15 + 10 + 10 = 80 (total students)