Introduction:
Electronegativity is a property of an atom that describes its ability to attract electrons towards itself in a chemical bond. It is an important concept in chemistry as it helps in understanding the nature of chemical bonds and the polarity of molecules. The periodic table provides a systematic arrangement of elements based on their properties, including electronegativity.

Explanation:
The electronegativity of elements generally follows a predictable trend across periods (horizontal rows) and groups (vertical columns) of the periodic table. Let's analyze the options provided and understand why option 'B' is the correct answer.

a) Increases across a period from left to right and increases down a group:
- Across a period: As we move from left to right across a period, the atomic number of elements increases, resulting in an increase in the number of protons in the nucleus. This increase in positive charge attracts electrons more strongly, leading to an increase in electronegativity.
- Down a group: Going down a group, the atomic size increases due to the addition of new energy levels. The increased distance between the nucleus and outermost electrons reduces the attractive force, resulting in a decrease in electronegativity. Therefore, this option is incorrect.

b) Increases across a period from left to right and decreases down a group:
- Across a period: As mentioned earlier, the atomic number increases across a period. This means that the effective nuclear charge (the positive charge experienced by an electron) also increases. As a result, the electrons are held more tightly, and the electronegativity of the elements also increases.
- Down a group: Going down a group, the atomic size increases, and the screening effect (outer electrons are shielded from the positive charge of the nucleus by inner electrons) becomes more significant. These factors reduce the effective nuclear charge on the outer electrons, leading to a decrease in electronegativity. Therefore, this option is correct.

c) Decreases across a period from left to right and increases down a group:
- Across a period: As discussed earlier, the electronegativity generally increases across a period from left to right. Therefore, this option is incorrect.
- Down a group: Going down a group, the atomic size increases, and the screening effect becomes more significant. These factors reduce the effective nuclear charge on the outer electrons, leading to a decrease in electronegativity. Therefore, this option is also incorrect.

d) Decreases across a period from left to right and decreases down a group:
- Across a period: The electronegativity generally increases across a period from left to right. Therefore, this option is incorrect.
- Down a group: Going down a group, the atomic size increases, and the screening effect becomes more significant. These factors reduce the effective nuclear charge on the outer electrons, leading to a decrease in electronegativity. Therefore, this option is also incorrect.

Conclusion:
Based on the analysis, it is clear that option 'B' is the correct answer. Electronegativity generally increases across a period from left to right and decreases down a group in the periodic table.