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Venn diagrams

Venn diagrams are today’s mostly used method for solving syllogisms. With some practice they can be drawn fairly quickly making them a valuable tool in solving syllogisms in timed aptitude tests.

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Venn diagrams show all possible and hypothetically logical relations between a collection of finite and infinite statements. By means of an overlap between some certain assumptions conclusions can be made using the (in)finite statements. Two examples of the use of Venn diagrams will follow to clarify the above.

Example 1:

  1. All Canadians are right handed
  2. All right handed are opticians
  3. Conclusion: Some opticians are Canadian

To check the validity of this statement first the different terms are appointed.

Subject:        Canadian
Predicate:      Optician
Middle term:    Right handed

We will start with the first out of the two given statements from above. The first thing to do is draw two circles and write the terms Canadian and Right handed in them. The circle with the word Canadian without the overlap represents only Canadian people, while the part within the overlap with the right handed circle represents all Right handed Canadian people. Everything outside these two circles represents everything not connected to these two terms. With this one can think of plants, animals, cars but even you and me.

Statement a 1 displayed as part of a Venn diagram.

1st Statement

Next, the 1st statement claims: all Canadians are right handed. Thus this means that all Canadian people outside the overlap of the two circles are not involved in this statement, since they are not connected to the term right handed. As a conclusion of that this part of the circle is being shaded.

2nd Statement

Subsequently the 2nd statement is reviewed. According to this statement all right handed are opticians. This statement can be solved by drawing two circles and again shading everything except the overlap in the right handed circle, just as was done with the first statement.

Linking Statements

Linking the two statements and the circles together results in the Venn Diagram of figure 2. Here both the first (red) as well as the second (green) statement are displayed. The overlap between Right handed and Optician is clearly shown, even as the absence of one between Canadian and Opticians. Further it can be noticed that there is a small area where all three term are overlapping, a part which is still present.

Venn diagram example 1

Now that the Venn diagram is completed, the validity of the conclusion can be checked. The conclusion states: some Opticians are Canadian. The Venn diagram clearly shows the correctness of this conclusion. Although the overlap area between both orange and green circle is shaded, there is still a small area in the middle where all three terms are present which it not shaded. It is this area that results in the correctness of the conclusion.

This case is characterized as a valid reasoning, since the conclusion can be drawn directly using the Venn diagram. It is however also possible that additional information is needed in order to check the validity of the conclusion. In that case the reasoning is invalid.

Example 2.

  1. All hamburgers are meals
  2. Some cows are hamburgers

Possible answers:

  1. All meals are cows
  2. At least some meals are cows
  3. No cows are meals
  4. Some cows are no meal

It is possible to assign a subject, predicate and middle term for all the statements. However, this would take lot of unnecessary time. Choosing between four statements when solving syllogisms can be handled best by making a Venn diagram straight away. In that way the possible answers from the statements can be checked on their validity piece by piece, resulting in the correct statement.

1st Statement

First statement 2a will be examined. The method behind drawing this part of the Venn diagram is exactly the same as the one explained in example 1, resulting in figure 3.

Fig. 3: Statement a 2 displayed as part of a Venn diagram.

In this way the first part of the Venn diagram displays that all hamburgers are meals, since the part with only hamburgers is shaded to result in the overlap area between the two terms; hamburgers and meals.

2nd Statement

Next statement 2b is examined. This statement needs a different approach since the statement claims the following: some cows are hamburgers. This means that it is not possible to just shade a whole area as was done before.

Statement b displayed as part of a Venn diagram.

In order to do that the statement should contain words like “all” or “none”. In this case the statement contains the word “some” and in that case a cross is used to represent that part of the statement in the Venn Diagram. Therefore a cross is put in the overlap between cows and hamburgers, representing the statement that some cows are hamburgers.

Linking Statements

Linking the two statements and the circles together results in the Venn Diagram of figure 5. With the help of this Venn diagram the 4 statements can be checked for their validity.

Venn diagram example 2

Checking Statements:

    1. All meals are cows.

However it can be seen that the term meals has an overlap with both hamburgers as well as cows, meaning that both are possible en thus resulting in an invalid statement.

    1. Some meals are cows.

This is correct, since the Venn diagram clearly shows a link between hamburgers and meals (a) and Cows and hamburgers (b). This automatically generates a link between meals and cows (be aware of the fact that there is no link between cows and meals). The Venn diagram clearly shows that this area is not shaded and thus a possible correct answer.

    1. No cows are meals.

It can easily be concluded that this statement is incorrect, since an overlap is present between these two terms.

    1. Some cows are no meals.

Be aware of the rank of the terms. It was already suggested that some cows are hamburgers, but nothing is stated between the relation of cows and meals. In statement 2 the rank was different so conclusions could be made, which in this situation is not the case.

In this example the correct answer is statement 2. Most syllogisms can be solved by using the above manner. The trick by solving syllogisms is often correct reading and interpreting of the statements and conclusions for obtaining a valid reasoning.

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