# Question #5604b

##### 2 Answers

They are logically equivalent; The detailed proof is given below.

#### Explanation:

In mathematical relations, **then** and **and** are represented by

So, we are to prove that either

or

**Note:** I have represented **equivalent sign** with **not equivalent** sign with

To find that either two statements are logically equivalent or not we use truth table. Before finding the truth tables, let me tell you the mathematical definitions of **then** and **and** relations.

**then:** A **then** statement of two statements **false** only when **true** but **false**, otherwise, it is always true. A **then** statement is known as **implication**.

**and:** An **and** statement of two statements **true** only when **true**, otherwise, it is always **false**. An **and** statement is known as **conjunction**.

Keep above definitions in mind.

In your given statement there are three basic statements

The basic three statements give rise to **rows** and **columns**.

In the first row, list all the statements as shown below.

Three basic statements give rise to 8 possibilities which can be listed in the table as shown below.

Now, fourth column represents **false** only when **true** but **false**. If you look into the table you will observe that **false** only for **false** only for

Now, sixth column represents **true** only when both **true**, otherwise **false.** In the table, **true** only in

For seventh column, you need to look in **and** operation on them, i.e, **conjunction** of **true** whenever both **true**. After looking into the table, I find that **false** for

Finally for eighth column, you need to look into **then** operation on them, i.e, **implication** of **false** only when **true**, but **false**. After looking into the table. I find that **false** for

After very good hard work, the truth table is ready.

If you look into

Hence, proved!

See below.

#### Explanation:

According to set theory

the second assertion is

and we know that

so it is true because if

This can be easily verified through a Venn graph.