Prove that 11x - 7 is even if and only if x is odd

TL;DR

Proving 11x - 7 is even if x is odd and showing the contrapositive where x being even makes it odd.

Transcript

hello in this problem we're going to do a proof we have that x is an element in the set of integers and we're going to prove that 11x minus 7 is even if and only if x is odd so before we do the proof let me refresh your memory on what it means for an integer to be even and odd because it's really important so recall that we say that x is even this ... Read More

Key Insights

• 🦕 Understanding the definition of even and odd integers is crucial in mathematical proofs.
• 🦕 Proofs involving even and odd properties require careful substitution and simplification.
• 👍 Utilizing contrapositive reasoning can simplify proving relationships between even and odd numbers.
• ☺️ The relationship between x being odd and 11x - 7 being even is a fundamental mathematical concept.
• ❓ Mathematical proofs often involve assuming certain conditions and deducing logical conclusions.
• #️⃣ Revisiting the basics of number properties is essential for tackling more complex proofs in mathematics.
• ❓ Errors in proofs can be easily rectified through careful review and correction.

Questions & Answers

Q: What does it mean for an integer to be even or odd?

An even integer is represented as 2k, while an odd integer is represented as 2k + 1, for some integer k.

Q: How is the proof structured to show that 11x - 7 is even if x is odd?

The proof starts by assuming x is odd, then substitutes and simplifies the expression to demonstrate that 11x - 7 is even.

Q: What approach is used to prove the contrapositive in the given scenario?

The contrapositive is proven by assuming x is even and then showing that 11x - 7 is odd, ultimately concluding the relationship between x's parity and the resulting expression.

Q: How does the proof confirm the relationship between x's parity and the even/odd nature of 11x - 7?

By carefully substituting the expressions for x, performing the necessary calculations, and ensuring that the resulting expression aligns with the definition of even and odd numbers.

Summary & Key Takeaways

• Introduces the concept of even and odd integers.

• Demonstrates a proof showing that 11x - 7 is even when x is odd.

• Utilizes contrapositive reasoning to prove that if 11x - 7 is even, then x is odd.