Proposition

Proposition is a statement that tells a fact despite of its exactness

Example:

 

He is Pang Kok An

He is pregnant.

1 + 1 = 2

2 + 3 = 6

The above examples are examples of propositions. The first statement states the fact that the person in the picture is Pang Kok An. It’s a true statements, therefore it’s a propositions. 

The second statement states that he is pregnant. General knowledge tells us that a man cannot be pregnant. Therefore, although it’s false but still be considered as a proposition because the statement had told a fact, although it’s a wrong fact.

Statement 3 tells the fact that 1 + 1 = 2. Therefore it’s a proposition. The same goes to Statement 4.

Let’s see some examples:

 Do you like peanuts?

 Please listen carefully!

 x + 1 = 2

 x + y = z

Statement 1 asks you do you like peanuts. You can say the sentence is neither true nor false. That is because you can determine its exactness since it doesn’t tell any fact. The same goes to Statement 2. Statement 2 does not tell you the fact but it’s a sentence that shows up a request.

x + 1 = 2 is also nor true nor false. If you put 1 as x, it would be true. But if you put

2 as x, the statement would turn wrong. Same goes to Statement 4.

The statements which are neither true nor false are not propositions.

Propositional variable

We have learnt before that a variable is something that represents a value.  

x = 2

The statement above shows that the variable x is representing the number 2.

A propositional variable is a variable which represents a proposition.

For example:

x = My lecturer is Madam Elissa

y = She is still singled.

The above example shows that variable x is representing the proposition “My lecturer is Madam Elissa” whereas the variable y represents the proposition “She is still singled”.

Truth Value

A true proposition has a truth value of T, whereas a false proposition has a truth value of  F. 

Negation

A negation to the proposition statement is what makes a positive proposition into negative and vice versa.

Example 1:

x = My lecturer is Madam Elissa

Turn this proposition into a negative statement and you will get:

¬x = My lecturer is not Madam Elissa/Madam Elissa is not my lecturer.

The symbol ¬ means not. Therefore ¬x means not x.

Example 2:

y = She is still singled.

¬y = She is not singled. / She is married.

x	¬x
T	F
F	T

Table above shows truth table for negation

Connective operators

Connective operators are used to connect two or more propositions and thus produce a new proposition. There are many types of connective operators.

AND operator

Connective operator “and” connects two propositions that will become a true compound proposition only if two of them are true.

Example:

x = My lecturer is Madam Elissa  

y = She is married.

There are 2 propositions. We use “and” to connect these 2 propositions and it will become

x ∧ y = My lecturer is Madam Elissa and she is married.

Note that the symbol ∧ represents the operator “and”.

In order to have the compound proposition true, the two propositions  must be true.

x	y	x ∧ y
T	T	T
T	F	F
F	T	F
F	F	F

Truth table for operator ∧

OR operator

Connective operator “or” connects two propositions that will become a true compound proposition if at least one of two of them are true.

Example:

x = She may have an egg

y = She may have a cup of coffee

There are 2 propositions. We use “or” to connect these 2 propositions and it will become

x ∨ y = She may have an egg or a cup of coffee.

Note that the symbol ∨ represents the operator “or”.

In order to have the compound proposition true, the two propositions  must be true.

x	y	x ∨ y
T	T	T
T	F	T
F	T	T
F	F	F

Truth table for operator ∨

Exclusive-OR operator

Connective operator “exclusive-or” connects two propositions that will become a true compound proposition if only one of two of them are true.

Example:

x = Those who take Akidah dan Akhlak in Sem 2 have registered their courses correctly.

y = Those who take Moral dan Etika in Sem 2 have registered their courses correctly.

x ⊕ y = For those who take Akidah dan  Akhlak or Moral dan Etika, BUT NOT BOTH have registered their courses correctly.

Note that the symbol ⊕ represents the operator “or”.

In order to have the compound proposition true, only one of the two propositions  must be true.

x	y	x ⊕ y
T	T	F
T	F	T
F	T	T
F	F	F

Truth table for operator ⊕

Conditional Statement

Conditional statement connects two proposition, one of them is the hypothesis part and the other one is the conclusion part. 

Example 1:

x = You are an IT student.

y = you must be familiar enough with computers.

Let x to be the hypothesis part and y as the conclusion part. Then it should be look like this:

x → y = If you are an IT student, you must be familiar enough with computers.

Note that the symbol → represents if…..then, so in this case, x → y means if x, then y.

The above statement is true. IT students must be familiar with computers.  

Example 2:

¬x →¬y = If you are not an IT student, you may not be familiar enough with computers.

The above statement is true. Non-IT students are not familiar with computers. 

Example 3:

¬x →y = If you are not an IT student, you may be familiar enough with computers.

The above statement is true. Some non-IT students are familiar with computers. 

Example 4:

x →¬y = If you are an IT student, you must not be familiar enough with computers.

The above statement is unreasonable. IT students have no reason why they are unfamiliar with computers.

All the results are true except for compound propositions with true hypothesis and false conclusion.

x	y	x ∨ y
T	T	T
T	F	F
F	T	T
F	F	T

Truth table for conditional statement

Biconditional Statement

Biconditional statement is denoted with double-arrow sign ↔. x ↔ y means x if and only if y.

Example 1:

x = It is not going to rain.

y = There are no grey clouds

1.  x ↔ y = It is not going to rain if and only if there are no grey clouds.

2. ¬x →¬y = It is not going to rain if and only if there are no grey clouds.

3. ¬x →y = It is not going to rain if and only if there are grey clouds.

4. ¬x →¬y =It is not going to rain if and only if there are no grey clouds.

Statement 1 and 4 are reasonable. Statement 2 and 3 are not reasonable because in order to rain, it needs grey clouds.

From the example above, you can see that if biconditional statement is true if two propositions are of same truth value.

x	y	x ∨ y
T	T	T
T	F	F
F	T	F
F	F	T

Truth table for biconditional statement