**Question**

**Introduction**

This is an
olympiad type of question that tests one’s use of logic. Sadly, logic is seldom taught in schools,
except maybe in courses like knowledge inquiry, theory of knowledge, or in selected topics like geometry. Mathematics
is actually very much connected with logic, but it seems to be regarded as
difficult to teach. Being able to use logic
and think critically is very important in our lives in various aspects.

Let us
study how we can analyse *conditional statements*using logic.

**Analysing Conditional Statement with Logic**

A

For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So

**is a statement of the form “If***conditional statement**H*then*C*.” An example would be: “If the blue litmus paper turns red, then the liquid is an acid”. The front clause*H*(the part about the litmus paper turning red) is called the**(or***hypothesis***). The latter clause***premise**C*(the part about the acid) is called the**. Every conditional statement has three other related forms: the contrapositive, the converse, and the inverse.***conclusion*For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old. So

*H*:= “watch NC16” and*C*:= “16 years old and above”. The**means “If you want to watch the movie, then you must be 16 years old and above”. The***conditional statement***is “If you below 16 years old, then you cannot watch the movie”. The***contrapositive**contrapositive*always has exactly the same meaning as the*original*conditional statement. If one is true, so is the other. If one is false, so is the other. The contrapositive is merely phrased in a different way.
The

The converse (and the inverse) has a different meaning
from the original statement. If the
original statment is true, the converse may or may not be true. In our
example, “If you want to watch the movie, then you must be 16 years old and
above” is true, but the converse “If you are 16 years old and above, then you
want to watch the movie.” may not be
true, as some people who are 16 years old and above may want to do something
else. Like reading a book. Or playing golf. So you see, there are really two camps: the **says “If you are 16 and above, then you will watch the movie.” The***converse***would say “If you don’t want to watch the movie, then you are not 16 and above.” Note that the inverse is the contrapositive of the converse. The***inverse**inverse*always has exactly the same meaning as the*converse*. If one is true, so is the other. If one is false, so is the other.*original conditional statement*and its*contrapositive*in one camp, and the*converse*and the*inverse*in the other.
Let us
analyse the given problem using logic.

**Analysis**

The
statement ‘any card with a letter

**A**on one side always has the number**“1”**on the other side’ can be rephrased as ‘If letter**A**appears, then the other side is**“1”**’. Let us analyse each position in order. For (a), since the letter**A**appears, we definitely need to know what is behind the card to test whether the claim is true. For (b), note that the claim does not say ‘If the letter is not**A**, then the other side is not**“1”**’ (*inverse*) nor ‘If the number is**“1”**, then the other side must be an**A**.’ (*converse*). It is possible that**A**is on the other side and we are fine. But it is also possible to have a non-**A**with the number**“1”**. The given condition does not prohibit that. So, if the other side is not an A, it is still OK. So the other side can be anything, and it does not matter. For (c) we definitely have to flip to check the other side in case the other side is an**A**, then the claim would be proven false. For (d), since the card is not an**A**, it is irrelevant. The claim is talking about**A**. It is not talking about**B**. We can summarise the analysis in the diagram below:-

**Ans:**We need to flip (a) and (c)

**Suitable Levels**

*****Primary School Olympiad

* syllabuses that involve logic and epistemology

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