I'll go through each question and provide step-by-step solutions for them.

Question 1

Given propositions:

• $p$: Adit is an exemplary student.
• $q$: Dita is an outstanding student.

You are asked to express the following propositions in simple sentences.

Solutions:

1. (a) $p \rightarrow q$: "If Adit is an exemplary student, then Dita is an outstanding student."
2. (b) $\neg q \land p$: "Dita is not an outstanding student, and Adit is an exemplary student."
3. (c) $\neg (p \leftrightarrow q)$: "It is not the case that Adit being exemplary is equivalent to Dita being outstanding."

Question 2

Consider the proposition:

"If the product of two numbers is odd, then both numbers are odd."

We need to determine the inverse, converse, and contrapositive of this proposition.

1. Inverse: Negate both the hypothesis and conclusion:

   "If the product of two numbers is not odd, then at least one of the numbers is not odd."

2. Converse: Swap the hypothesis and conclusion:

   "If both numbers are odd, then the product of the two numbers is odd."

3. Contrapositive: Swap and negate both hypothesis and conclusion:

   "If at least one of the numbers is not odd, then the product of the two numbers is not odd."

Question 3

Complete the truth table given the operation $\Delta$ defined in the following ways:

1. Truth Table Columns:
   • $p$
   • $q$
   • $\neg q \Delta p$
   • $\sim p \Delta q$
   • $\sim p$

The missing cells can