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Discrete Mathematics

Let ๐‘ƒ(๐‘ฅ) and ๐‘„(๐‘ฅ) be predicates and suppose ๐ท is the domain of ๐‘ฅ. For the given pair of statement forms determine whether (a) the two forms have the same truth value for every choice of ๐‘ƒ(๐‘ฅ),๐‘„(๐‘ฅ), and ๐ท, or (b) there is a choice of ๐‘ƒ(๐‘ฅ),๐‘„(๐‘ฅ), and ๐ท for which the forms have opposite truth values.

โˆƒ๐‘ฅ โˆˆ ๐ท | (๐‘ƒ(๐‘ฅ) โˆง ๐‘„(๐‘ฅ)) ๐‘ฃ๐‘’๐‘Ÿ๐‘ ๐‘ข๐‘  (โˆƒ๐‘ฅ โˆˆ ๐ท | ๐‘ƒ(๐‘ฅ)) โˆง (โˆƒ๐‘ฅ โˆˆ ๐ท | ๐‘„(๐‘ฅ))

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