Question

Which of the following connectives satisfy commutative law?
A. $\wedge $
B. $\vee $
C. $\Leftrightarrow $
D. All of the above

Answer 1

Hint: First we will analyze all the types of connectives and then we will write down the commutative law. After that we will see for each option that whether or not they satisfy the commutative law and then we will get the answer.

Complete step by step answer:
Now we know that logical connectives or logical operators are basically functions of one or more variables, where the variables can be either True or False and the value of the function can be True or False. There are commonly five types of connectives used in mathematics which are as follows:
(i) Negation: The negation of a statement X means that the statement X is not true. The symbol of negation is \[\neg \]. So, you can write \[\neg X\] for Not $X$.
(ii) Conjunction: The conjunction of two statements X and Y means that both the statements are True. The logical symbol for conjunction is $\wedge $ . So, you can write X $\wedge $ Y for X and Y.
(iii) Disjunction: The disjunction of two statements X and Y means that at least one of the statements is True. The logical symbol for disjunction is $\vee $. So, you can write X $\vee $ Y for X OR Y.
(iv) Implication: The implication of two statements X and Y means that Y is True whenever X will be True. The logical symbol for implication is $\Rightarrow $. So, you can write X $\Rightarrow $ Y for X implies Y.
(v) Equivalence: The equivalence of two statements X and Y means that Y is True whenever X will be True and X will be true whenever Y is True. Another way of saying this is that X implies Y and Y implies X. The logical symbol for equivalence is $\Leftrightarrow $. So, you can write X $\Leftrightarrow $ Y for X iff Y.

Now we know that commutative law states the numbers or statements on which we operate can be moved around and it will not make any difference to the answer.
Now that we know about the connectives and commutative law we will take each option and find out which one satisfies the commutative law:
First option is $\wedge $ : Conjunction means AND. We see that here both statements need to be true. $\therefore x\wedge y=y\wedge x$ . Hence (i) satisfy the commutative law.
Second option is $\vee $ : Disjunction means OR. We see that at least one statement need to be true. $\therefore x\vee y=y\vee x$ . Hence (ii) satisfy the commutative law.
Third option is \[\Leftrightarrow \] : Equivalence means iff. We see that statement x will hold only and only if statement y is true. $\therefore x\Leftrightarrow y=y\Leftrightarrow x$ . Hence (iii) satisfy the commutative law.
We see that all three options satisfy the commutative law.

So, the correct answer is “Option D”.

Note: You need not write the whole definition of all the connectives, you can simply define the asked connectives and apply the commutative law and see if they satisfy it or not. The Implication can cause confusion in understanding; it means that statement Y will hold truth whenever X is true that does not apply vice-a-versa.