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Critical Thinking and Logic in Mathematics

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  • 0:05 Logic
  • 1:24 Propositions
  • 2:22 True or False
  • 2:57 Critical Thinking
  • 3:45 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Logic has its own unique language and way of defining what is true and false. Watch this video lesson to learn how you can critically think in the language of logic while working with math.

Logic

If Judy likes all things round, then Judy will love donuts.

This is a logical statement. Logic is the study of how to critically think about propositions or statements that are either true or false. The statement I just made about Judy came about from thinking critically about the proposition that Judy likes all things round and about donuts. I know the proposition that Judy likes all things round is true, and I know the proposition that donuts are round is true, as well. Because both propositions are true, I can link them together to reach the conclusion that Judy will love donuts because donuts are round. This is the way logic works.

Logic is very useful in the world of mathematics. Mathematicians use logic all the time to prove theorems and other mathematical facts. Everything we know about math right now is based off of these logical proofs. Without these, we wouldn't have our formulas, like the wonderful quadratic formula or the very useful Pythagorean Theorem.

Using logic in math is about mixing the specific language used in logic with the specific symbols used in math. Let me show you.

Propositions

In logic, propositions are simple statements that can either be true or false. Your propositions don't have to be complicated. They can be short ones like, 'All squares are yellow,' or 'Judy likes all things pink.' Your proposition is any statement that can be labeled as either true or false.

Logic propositions in math usually include math symbols. In geometry, you can have a proposition that says, 'Line AB is the bisector of line CD' with the corresponding math symbol for lines instead of the word 'line.' In algebra, your proposition can be as simple as x = 2. Depending on what kind of math you're working with, you can have a mixture of words with math symbols or all math symbols. What matters most is that your logic proposition can be labeled as either true or false.

True or False

Usually, your problem will tell you whether a statement is true or false. One thing to keep in mind here is that if your problem says something is true, you have to believe that. Don't over think the statement. If you see a statement such as 2 + 2 = 5, and the problem says that it is true, then you have to believe that and work with it, but only for that problem. I know it might be hard to do, but what is true and false in logic does not have to make sense in the real world.

Let's now see how we can apply logic and critical thinking to a problem.

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