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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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.
Fred
4.4112 Answers
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.
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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