Logical deduction with biconditional (↔): Statement: “If and only if it is July, it will rain.” Consider the four simple statements: (i) It rains. (ii) It does not rain. (iii) It is July. (iv) It is not July. Which option lists a pair of statements that can be true simultaneously under the given biconditional?

Introduction / Context:
Biconditional reasoning (“if and only if”) links two propositions so that each is both necessary and sufficient for the other. Here, the sentence “If and only if it is July, it will rain” asserts an exact pairing between the month being July and rain occurring. We must determine which pair of simple statements can be true at the same time without contradicting this biconditional.

Given Data / Assumptions:

• Let J represent “It is July.”
• Let R represent “It rains.”
• The given statement is J ↔ R (J if and only if R).
• Simple statements listed are: (i) R, (ii) not R, (iii) J, (iv) not J.

Concept / Approach:
For a biconditional J ↔ R, the truth table has two satisfying rows: both true (J = true, R = true) or both false (J = false, R = false). Mixed cases (J true and R false, or J false and R true) violate the biconditional. Therefore, the only compatible pairs among the options are the ones where J and R have the same truth value.

Step-by-Step Solution:

Verification / Alternative check:
If there were an option listing (ii) (iv) (i.e., not R and not J), that would also be valid. Since it does not appear, the only correct pair provided is (i) with (iii).

Why Other Options Are Wrong:

• (i) (ii): Contradictory—cannot both rain and not rain.
• (iii) (ii) / (ii) (iii): These encode J and not R, which contradicts J ↔ R.
• None of these: Not appropriate because (i) (iii) is valid.

Common Pitfalls:

• Reading “if and only if” as a one-way implication; it is a two-way link.
• Assuming “if July then rain” but allowing “rain without July.” The biconditional forbids that.

Final Answer:

(i) (iii)