Logical Operators

1) Negation (~, 𝑛𝑜𝑡)

Let ‘𝑝’ be a proposition. The Negation of ‘p’ denoted by "~𝑝".

𝑇ℎ𝑒 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 “~𝑝” 𝑖𝑠 𝑡ℎ𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 ‘𝑝’

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝑁𝑒𝑔𝑎𝑡𝑖𝑜𝑛

𝑝 ~𝑝

1 0

0 1

2) Conjunction (⋀, 𝑎𝑛𝑑)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The conjunction of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ⋀ 𝑞".

"𝑇ℎ𝑒 𝐶𝑜𝑛𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 "𝑝 ⋀ 𝑞" 𝑖𝑠 𝟏 𝑤ℎ𝑒𝑛 𝑏𝑜𝑡ℎ ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ 𝑎𝑟𝑒 𝟏 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟎"

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝐶𝑜𝑛𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛

𝑝 𝑞 𝑝 ⋀ 𝑞 1 1 1

1 0 0 0 1 0

0 0 0

3) Disjunction (⋁, 𝑜𝑟)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The disjunction of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ⋁ 𝑞".

"𝑇ℎ𝑒 𝐷𝑖𝑠𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 "𝑝 ⋁ 𝑞" 𝑖𝑠 𝟏 𝑤ℎ𝑒𝑛 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑜𝑓 ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ 𝑖𝑠 𝟏 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟎"

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝐷𝑖𝑠𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛

𝑝 𝑞 𝑝 ⋁ 𝑞 1 1 1

1 0 1 0 1 1

0 0 0

4) Implication (Conditional) (→, 𝑖𝑚𝑝𝑙𝑖𝑒𝑠)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The conditional statement of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 → 𝑞".

"𝑇ℎ𝑒 𝐶𝑜𝑛𝑑𝑖𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑆𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 "𝑝 → 𝑞" 𝑖𝑠 𝟎 𝑤ℎ𝑒𝑛 ‘𝑝’ 𝑖𝑠 𝟏 𝑎𝑛𝑑 ‘𝑞’ 𝑖𝑠 𝟎 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟏"

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝐼𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛

𝑝 𝑞 𝑝 → 𝑞 1 1 1

1 0 0 0 1 1

0 0 1

5) Bi-implication (bi-conditional) (↔, 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The bi-conditional statement of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ↔ 𝑞".

"𝑇ℎ𝑒 𝐵𝑖 − 𝑐𝑜𝑛𝑑𝑖𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑆𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 "𝑝 ↔ 𝑞" 𝑖𝑠 𝟏 𝑤ℎ𝑒𝑛 𝑏𝑜𝑡ℎ ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ ℎ𝑎𝑣𝑒 𝑠𝑎𝑚𝑒 𝑣𝑎𝑙𝑢𝑒

𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟎"

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝐵𝑖 − 𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛

𝑝 𝑞 𝑝 ↔ 𝑞 1 1 1

1 0 0 0 1 0

0 0 1

6) Exclusive or (⨁, 𝑒𝑥 𝑜𝑟)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The exclusive or of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ⨁ 𝑞".

𝑇ℎ𝑒 𝐸𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 𝑜𝑟 𝑝 ⨁ 𝑞" 𝑖𝑠 𝟏 𝑤ℎ𝑒𝑛 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 𝑜𝑛𝑒 𝑜𝑓 ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ 𝑖𝑠 𝟏 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟎"

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝐸𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 𝑜𝑟

𝑝 𝑞 𝑝 ⨁ 𝑞 1 1 0

1 0 1 0 1 1

0 0 0

7) NAND Logical Operator (↑)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The NAND of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ↑ 𝑞".

𝑇ℎ𝑒 𝑁𝐴𝑁𝐷 "p↑q" 𝑖𝑠 𝟎 𝑖𝑓 𝑏𝑜𝑡ℎ 𝑜𝑓 𝑖𝑡𝑠 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟𝑠 ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ 𝑎𝑟𝑒 𝟏 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟏

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝑁𝐴𝑁𝐷

𝑝 𝑞 𝑝 ↑ 𝑞 1 1 0

1 0 1 0 1 1

0 0 1

8) NOR Logical Operator (↓)

Let ‘𝑝’ and ‘𝑞’ b two propositions. The Nor of ‘𝑝’ and ‘𝑞’ denoted by "𝑝 ↓ 𝑞".

𝑇ℎ𝑒 𝑁𝑜𝑟 "p↓q" 𝑖𝑠 𝟏 𝑖𝑓𝑏𝑜𝑡ℎ 𝑜𝑓 𝑖𝑡𝑠 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟𝑠 ‘𝑝’ 𝑎𝑛𝑑 ‘𝑞’ 𝑎𝑒 𝟎 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟏

𝑇𝑟𝑢𝑡ℎ 𝑇𝑎𝑏𝑙𝑒 𝑜𝑓 𝑁𝑜𝑟

𝑝 𝑞 𝑝 ↓ 𝑞 1 1 0

1 0 0 0 1 0

0 0 1