Back To CourseMath 106: Contemporary Math
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Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.
Have you ever heard the statement, 'Those two go together like peanut butter and jelly,' or 'You never see Mary without Jane'? These are common examples of connective reasoning.
To understand connective reasoning, you must have a slight understanding of sentence structure. You can have simple statements, which are statements that contain just one idea, like, 'I like dogs.' There are also compound statements. These statements are statements that have more than one idea combined through the use of logical operations, like, 'I like dogs and cats, but not fish.' In this compound statement, dogs and cats are connected through the operation AND; dogs and fish are effectively disconnected through the operation NOT.
In this lesson, we will learn the five main logical connectives through definitions and examples. The five connectives introduced in this lesson will be negation, conjunction, disjunction, conditional, and biconditional. Let's get started.
Negation is really just what it sounds like: negative or not. The sign for negation can be found on your keyboard just before the numeral 1 key. It looks like this: ~. Any time you see that symbol in a connective scenario (be it math or otherwise) you are determining what is NOT true.
Take our earlier compound statement example. The person likes dogs and cats, but not fish. This connective logic is negation because the fish are connected to the person by not being within the group of animals enjoyed.
Venn diagrams are a great way to understand all connective language. These diagrams are a visual representation of what items are in which group or set. In this Venn diagram, you can see two circles that are crossed over each other. One is labeled A, the other B. The place where they cross over contains items from both A and B. Anything not in either of these circles would be a negation. So, we can imagine that fish, in our example, would not be within either circle, thus NOT fish (~fish).
Venn diagrams are very common in statistical reporting to show numbers of separate groups that have common characteristics.
The next connective is conjunction. Conjunction refers to the concept of AND. Think about it this way: CON = CONnect, which means together. If you had a two train cars, they are just two train cars until you CONnect them together. Once they are connected together, you have: 'This car AND that car together.' You must have both cars to form the conjunction.
It is the same in mathematical reasoning. A conjunction refers to more than one subject of a statement needing to be true for the statement to the true. In our Venn diagram, to have a conjunction we must have both dogs AND cats. The symbol for conjunction is an upside-down V as shown in the diagram.
Alternatively, disjunction refers to the concept of OR. If the CON in conjunction stands for connection, then the DIS in DISjunction stands for DISconnect. We can have either the first car or the second car to have a train.
The interesting thing in connective reasoning is that a disjunction, the OR concept, can also be true if you have both items. So, it means 'either, or, or both.' You don't have to have both, but if you do, you still have satisfied the reasoning statement logically. The symbol for disjunction is V.
It is pretty straight forward to understand that CONjunction = CONnect and DISjunction = DISconncect. But, how can you remember the symbols that go with each? It is simple if you remember what each means. The symbol for conjunction is an upside-down V, it looks a bit like a tepee. The operation is AND; two things must be together to satisfy the logic. Think about the things (like dogs and cats) being trapped inside the tepee.
Conversely, the symbol for disjunction is a right-side-up V and it means OR; you don't need both things, but you could have them and still be okay. Visualize our dogs and cats coming and going at will. The only constraint is that you must have at least one animal in the V shape at any given time; other than that, they are free to move around. If you just visualize those two scenarios, you should be able to figure out which symbol you need for which type of connective reasoning.
Conditional connectives imply an IF --> THEN statement. This means that IF you have one thing, THEN you must also have another defined thing. With our animal example, 'IF you have animals, THEN you must have animal food,' is the connective reasoning we might use to explain why someone would have dog food in their house.
This is used in math exactly the same way: 'IF 2p --> THEN the product is an even number.' This connective reasoning statement is true because we know that any number multiplied by two gives a result that is even.
I'm sure you noticed that the symbol for a conditional statement is an arrow pointing at the result (-->).
The final connective reasoning statement we will look at is the biconditional statement. This is an IF --> THEN statement that works in both directions.
If we think about the conditional statement we had previously, 'IF you have animals, THEN you have animal food,' we can easily see that it does not necessarily work in reverse, 'IF you have animal food, THEN you have animals.' There are many reasons why a person could have animal food in the house but not actually own an animal (maybe their dog died and they haven't gotten rid of the food yet, or maybe they are taking care of a neighbor's pet and keeping the food close by... who knows?).
For biconditional statements to be true, the statement must work in both directions, thus the symbol is a two sided arrow, like this: < -- >.
Consider this statement: 'IF a triangle is equilateral, THEN all its sides are equal.' This is true. However, it is also true to say, 'IF a triangle has all sides equal, THEN it is an equilateral triangle.' See, both directions of the statement are true, thus the symbol is a bidirectional arrow (remember, 'bi-' means 2, so both directions).
So, there you have it, the five main logical connectives:
I hope this helps you understand connective reasoning better. Thanks for watching.
After this lesson is finished, you should be able to identify and describe the five most common logical connectives in math.
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Back To CourseMath 106: Contemporary Math
9 chapters | 106 lessons