Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.

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Answer to a math question Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.

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1. Identify the implications of the statement "x is an even integer if and only if 5x+3 is an odd integer." This is a biconditional statement, which means it can be expressed as two conditional statements (implications).

2. Break down the biconditional statement into its two conditional components:

- Conditional 1: If $ x $ is an even integer (hypothesis), then $ 5x+3 $ is an odd integer (conclusion).

- Conditional 2: If $ 5x+3 $ is an odd integer (hypothesis), then $ x $ is an even integer (conclusion).

3. Combine the two conditionals with "and" to form the conjunction of the conditionals:

- "If $ x $ is an even integer, then $ 5x+3 $ is an odd integer" $ \land $ "If $ 5x+3 $ is an odd integer, then $ x $ is an even integer."

**Answer:**

If $ x $ is an even integer, then $ 5x+3 $ is an odd integer.

If $ 5x+3 $ is an odd integer, then $ x $ is an even integer.

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Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.

Answer to a math question Suppose x is an integer. Consider the proposition, “x is an even integer if and only if 5x+3 is an odd integer.” Express this proposition as a conjunction of two distinct conditional statements.

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