What is Venn Diagram? Proof the Validity/Invalidity of the Following Using the Venn Diagram Method:
a) Some pens are pencils. Some pencils are blue. Therefore some pens are blue.
b) No scientist is a musician. All musicians are poets. Therefore, Some Scientists are poets.
c) All dogs are animals. Some animals are wild animals. Therefore some dogs are not wild animals.
Some pens are pencils. Some pencils are blue. Therefore some pens are blue.
A Venn diagram is a graphical representation of the relationships between sets. It is named after John Wayne, who introduced the concept in the 1880s. A Venn diagram consists of two or more circles that overlap to represent the intersection of sets. A circle represents the set of all pencils, and another circle represents the set of all pens.
To prove the validity or invalidity of the statement “Pens are pencils. Some pencils are blue. Therefore some pens are blue,” we can use a Venn diagram as follows:
We start by drawing two circles, one representing the set of all pencils and the other representing the set of all pens. Since “pens are pencils,” the circle representing the set of pens is a subset of the circle representing the set of pencils. This means that all pens are also pencils.
Next, we add the statement “Some pencils are blue” to the Venn diagram. This means that there is a subset of pencils which are blue. We can represent this subset by shading in a part of the circle representing the set of pencils.
Finally, we can use the information in the Venn diagram to determine whether the statement “Some pens are blue” is true or false. Since all pens are also pencils, and some pencils are blue, it follows that some pens must also be blue. Therefore, the statement “Some pens are blue” is true.
We can also use Venn diagram to prove the invalidity of the statement “Some pens are blue. Therefore, pens are pencils.” To do this, we’ll draw a Venn diagram with two circles, one representing the set of pencils and the other representing the set of pens. Since “some pens are blue,” we will shade in a portion of the circle representing the set of pens. However, this does not mean that all pens are pencils. There may be pens that are not blue and therefore are not included in the subset of blue pens. Therefore, the statement “pens are pencils” is not necessarily true, and the argument is invalid.
Finally, Venn diagrams are a valuable tool for visualizing and proving the validity or invalidity of statements involving sets. By representing the relationship between sets with overlapping circles, we can easily see the effect of one statement on another, and determine whether an argument is valid or invalid.
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