I have gotten the first three done but need help with the last two please show work. Step 1Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three sub-partsfor you. To get remaining sub-parts solved please repost the complete question and mention the sub-parts to be solved.The logical symbol "v" is used to denote "OR". The logical symbol "-" is used to denote "NOT".Any two propositions are said to be logically equivalent if the truth value in each row of the truth table isthe same for both propositions.(a)It is provided that P↓Q is true if both P, Q are false and false in all other scenarios. Use this data to makethe truth table of P↓Q as follows.PQTTTFFTFFFTMake the truth table of ¬PvQ as follows.PQPVQTPTTTTFTFTודFFFFFTComparing the last columns of both truth tables, it can be concluded that the expressions: P+Q andPvQ are logically equivalent.Q(b)Make the truth table of QvP as follows.PQTTTFFFComparing the last columns of both truth tables, it can be concluded that the expressions: P Q and QvPare logically equivalent. Hence, is commutative.TTFTFLFTTTF(c)Make the truth table of P Q R and P Q R as follows.P↓↓RTFRTFTFTFTP+QFFFLLP↓QFFFFTFTTQ+PFFFTFTFT-PVQFTFQ&RFFFTFFFFFFP↓QRFFFFTTTFFFTFTFComparing the columns of P Q R and P Q R, it can be concluded that the expressions: P+Q+R andP Q R are logically not equivalent. Hence, it is proved that is not associative. 1. We define a logical connective ↓ as follows: P↓ Q is true when both P and Q are false,and it is false otherwise. (We read P↓ Q as "P nor Q”).(a) Write a truth table for P↓ Q and check that P↓ Q is logically equivalent to¬(P V Q).(b) Check that P↓ Q = Q ↓ P. That is, is commutative.(c) Show that (P ↓ Q) ↓ R and P ↓ (Q ↓ R) are logically inequivalent. That is, ↓ isnot associative.(d) Show that the logical connectives ¬, A, and V can each be expressed entirely interms of ↓, without using any other logical connectives. Specifically, prove thefollowing:i. ¬P = (P ↓ P).ii. P^Q = (P ↓ P) ↓ (Q ↓ Q).iii. PVQ (P ↓Q) ↓ (P↓ Q).(e) Prove that the logical connective ⇒ can be expressed entirely in terms of ↓. Thatis, show that the sentence P⇒ Q is logically equivalent to a sentence involving ↓and no other logical connectives.

Question

I have gotten the first three done but need help with the last two please show work. Step 1Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three sub-partsfor you. To get remaining sub-parts solved please repost the complete question and mention the sub-parts to be solved.The logical symbol &#34;v&#34; is used to denote &#34;OR&#34;. The logical symbol &#34;-&#34; is used to denote &#34;NOT&#34;.Any two propositions are said to be logically equivalent if the truth value in each row of the truth table isthe same for both propositions.(a)It is provided that P↓Q is true if both P, Q are false and false in all other scenarios. Use this data to makethe truth table of P↓Q as follows.PQTTTFFTFFFTMake the truth table of ¬PvQ as follows.PQPVQTPTTTTFTFTודFFFFFTComparing the last columns of both truth tables, it can be concluded that the expressions: P+Q andPvQ are logically equivalent.Q(b)Make the truth table of QvP as follows.PQTTTFFFComparing the last columns of both truth tables, it can be concluded that the expressions: P Q and QvPare logically equivalent. Hence, is commutative.TTFTFLFTTTF(c)Make the truth table of P Q R and P Q R as follows.P↓↓RTFRTFTFTFTP+QFFFLLP↓QFFFFTFTTQ+PFFFTFTFT-PVQFTFQ&RFFFTFFFFFFP↓QRFFFFTTTFFFTFTFComparing the columns of P Q R and P Q R, it can be concluded that the expressions: P+Q+R andP Q R are logically not equivalent. Hence, it is proved that is not associative. 1. We define a logical connective ↓ as follows: P↓ Q is true when both P and Q are false,and it is false otherwise. (We read P↓ Q as &#34;P nor Q”).(a) Write a truth table for P↓ Q and check that P↓ Q is logically equivalent to¬(P V Q).(b) Check that P↓ Q = Q ↓ P. That is, is commutative.(c) Show that (P ↓ Q) ↓ R and P ↓ (Q ↓ R) are logically inequivalent. That is, ↓ isnot associative.(d) Show that the logical connectives ¬, A, and V can each be expressed entirely interms of ↓, without using any other logical connectives. Specifically, prove thefollowing:i. ¬P = (P ↓ P).ii. P^Q = (P ↓ P) ↓ (Q ↓ Q).iii. PVQ (P ↓Q) ↓ (P↓ Q).(e) Prove that the logical connective ⇒ can be expressed entirely in terms of ↓. Thatis, show that the sentence P⇒ Q is logically equivalent to a sentence involving ↓and no other logical connectives.

Accepted Answer

Given:  is true if both operands are false and false in every other scenario. (d) To show the…